No, there was never such a period. The US has always had border control and standards for who it let in. It was just that those standards used to be extremely low. Prior to the 1870s, almost everyone who applied were given resident status and became citizens after a set period. The lowest estimate for that period is 90% of people who applied. During the Progressive period, increasing restrictions came about, decreasing the admittance rate. Then comes 1924, when a temporary act from 1921 is made permanent. That is the birth of immigration enforcement offices, the immigration courts, quotas, deportation, and basically everything about our modern immigration system.

This was a culmination of two growing trends in Americans politics: progressivism and racism/nativism. The laissez-faire economics, racial reconstruction, and free immigration of the Gilded Age gave way to interventionism, Jim Crow, and the inspection system of the Progressive Era. These were entirely consonant at the time. Labor unions often discriminated against nonwhite members and campaigned hard for immigration restrictions. Progressive academics began espousing eugenic theories as well as what we now call scientific racism. Progressivism was fundamentally about improving society, and that meant things like welfare, a living wage, 'managing' the economy to make it more efficient, and so on. But it also meant the forced absorption of minorities (like Native Americans) and the limiting or reducing of genetically inferior populations (like the Italians or the African Americans). The Progressives thought they could improve everything, including human beings themselves, through this sort of management. In addition, there was a certain plain reactionary element. Nonwhite, non-protestant populations had exploded in the US and this led to an increase in (always existent) bigotry.

In line with these desires, the Federal government took greater control of immigration. It set up bureaucracies (as it was doing for many things) and opened checkpoints. It established tests for things like intelligence. It banned entire categories, such as the Chinese, people belonging to certain political parties, or anyone found to have ever engaged in prostitution. Various states also increased the disabilities of not being a citizen, such as forbidding land ownership. But while this did have a notable effect on immigration, the supermajority of people were still let in until 1924. The National Origins system was then put in place, explicitly with the goal of excluding East Asians and Southern Europeans, and encouraging northern Europeans.

Immigration decreased somewhat immediately. It decreased even further after the Great Depression. In fact it cratered. There was a higher rate of immigration during the Civil War than during the Great Depression. After 1932, immigration to the US effectively became impossible. But more than that, the Great Depression stoked all sorts discriminatory animus. Crowds chanted "no jobs for niggers" in WPA lines. The programs were often structured specifically to discriminate against various groups. Redlining, for example, had its origins in this period and had four grades, of which only the first suffered no discrimination.

This is when you start seeing mass deportation and immigration sweeps. Deportation was originally a very limited tool. But by the 1930s it was being used to deport hundreds of thousands at a time. It got so zealous that the US government eventually realized it had deported hundreds of thousands of American citizens, mainly Hispanics. While this was a minor scandal, some defenders of the policy simply labeled them as foreigners and moved on. And indeed, in society generally, coercing and being unwelcoming to foreigners was increasingly common. Some groups tried to force foreign born citizens out of the United States by not hiring them, for example. This declined in the patriotic fervor of World War 2, with some concessions coming as early as 1939.

Notably, some of this opposition ran all the way back to 1924. Emanuel Celler (a Democrat who was ahead of his time and his party in many ways) voted against the 1924 Act and was a key defender of immigrant rights. In the aftermath of the war, in 1952, a much more liberal act was passed. It made the quota system much fairer and less discriminatory, and the courts much less harsh, but did re-establish them. The national origin quotas were abolished in 1965, replaced by quotas on types of immigrants. Illegal immigration remained a legal reality but there weren't many of them. Foreign born percentage held relatively steady and relatively low. There were a few changes in the 1980s, mostly related to refugees. But that system still basically exists today.

Illegal immigration began to pick up a little in the late 70s and 80s. The increasing illegal population quickly became a political issue, which led to an amnesty and a 1986 act to specifically discourage illegal immigration. It enjoyed largely bipartisan support and was somewhat effective immediately, reducing what was already a relatively small population and decreasing the flow.

Then the 1990s happened. 1990-2000 is the greatest single decade increase of foreign born Americans in history. The foreign born population boomed and foreign born populations quickly reached historic highs, especially from Asia and Latin America. Illegal immigration exploded and became a wedge issue between the Republicans and Democrats, as it is today, with the first partisan line votes beginning around 1996.

TL;DR: Illegal immigration as a matter of law has always been possible. Border checkpoints and turning people away dates to the 1880s. Deportation and immigration courts date back to 1924. Illegal immigration as a major political issue dates back to the mid-nineties.

Follow up, if allowed, was the "invention" of illegal immigrants in any way tied to social programs like welfare and social security?

I won't get into pre-Colombian migration patterns in North America, because that's not a separate answer, that's an entire separate subfield. The short answer to your question: it depends what you mean by "illegal" and by "could arrive." From its earliest dates, European immigration to the Americas was limited as much by the resources commanded by potential immigrants as by legal conventions.

Since you mentioned U.S., I'm going to start with the British colonies in North America, as distinct from New France (Canada) and New Spain and New Portugal (Florida, Texas, most of California, the Caribbean, South America). Strictly speaking, the first illegal immigrants to the US-- immigrants who arrived in direct violation of a legal document ratified within the system of Western law-- were the settlers of Jamestown. The 1494 Treaty of Torsedillas, ratified by the Pope, had settled disputes between the monarchs of Spain and Portugal regarding rights to discover and settle the new continent, establishing a neat line that separated zones of Spanish influence from zones of Portuguese influence. As long as each of the bickering nations stayed on their side of the Pope-enforced line, there would be no need for war.

However, the political and religious situation changed significantly between 1494 and 1607. The Reformation and series of religious wars significantly weakened both the Vatican's legitimacy and influence, and the ability of Spain and Portugal to defend their territories in North America. England, France, and the Netherlands saw a golden opportunity. They claimed that since the Treaty of Torsedillas was a document signed by two Roman Catholic nations and ratified by the Pope, and they themselves were Protestant, the rules didn't actually apply to them. And since neither Portugal nor Spain could defend all of the new continent, England and France decided to focus on areas far to the north of Spanish settlements.

Jamestown was followed by a series of settlements in New England, the Thirteen Colonies. While each of the thirteen was founded by different groups of people with differing ideas about government, religion, and tolerance, the legal rights to the land followed the same pattern. Drawing from the British feudal legal tradition, the Crown would issue a charter to an individual or company. Officially, all land belonged to the British Crown, or was controlled by the Crown-- a distinction emerging in the evolving early modern legal system. The charter granted rights to farm, settle, and govern a particular tract of the land. These charters established a governor who represented the king and tasked him with upholding the Crown's immigration policy based around two policies. It's established in the 1606 Virginia Charter and recurs, but I've picked the 1629 Charter of the Massachusetts Bay to quote from because the language is particularly strong to govern upstart, anti-monarchical Puritans.

"That all and every the Subjects of Us... which shall goe to and inhabite within the saide Landes and Premisses hereby mentioned to be graunted, and every of their Children which shall happen to be borne there, or on the Seas in goeing thither, or retorning from thence, shall have and enjoy all liberties and Immunities of free and naturall Subjects within any of the Domynions of Us... as if they and everie of them were borne within the Realme of England. And that the Governor... shall and maie at all Tymes... have full Power and Authoritie to minister and give the Oathe and Oathes of Supremacie and Allegiance... to all and everie Person and Persons"

In this legal model, the land is controlled by a monarch (the King of England) and occupation of that land is contingent on allegiance to him. The Governor represents the King and has two jobs: 1) ensure that subjecthood is extended to English subjects born in the Americas 2) make sure they don't forget to be loyal to the King. As you know, these two legal principles did not always succeed in shaping political realities in the colonies.

The first problem was that many of the religious immigrants were technically English subjects, but were Puritans lately involved in the English Civil War who'd fled to Holland afterwards. They did not like the king; they didn't like the concept of a king; and they tended to set up parallel governments with documents like the Mayflower Compact. Or would subvert the legal form, setting up a government had a Governor as described in the charter but ran along more democratic lines. Or more authoritarian lines, but with closer resemblance theocracy than monarchy. Meanwhile in Pennsylvania, William Penn gets a much less circumscribed charter from Charles II because the Crown owed Penn's dad a lot of money. Penn embarks on the "Holy Experiment" and surrenders his privileges of government to assembly, institutes principles of religious toleration, and completely omits oaths of allegiance. The Assembly shall ensure that all immigrants "shall have all other Powers and Privileges of an Assembly, according to the Rights of the free-born Subjects of England, and as is usual in any of the King's Plantations in America." Things get even more complicated when the British colonies expand after the Seven Years' War and include colonies along the Atlantic seaboard originally founded by the Dutch and Swedish.

As the seventeenth century turned into the eighteenth, the legal forms of government in the British colonies looked less and less like the monarchical prescriptions of the charters. The realities of immigration were also changing. As Aaron Spencer Fogleman has demonstrated, an estimated half-million people immigrated to the North American British colonies between 1650-1775. The vast majority were not British: nearly half were Africans, enduring forced migration to American plantations; one-fifth were from Ireland (although many were Anglo-Irish); and one-seventh were German speakers from current-day Germany, Switzerland, and France. How did so many different nationalities arrive in the colonies, despite the charters' clear regulation of His Majesty's subjects? The answer is indentured servitude, driven by a need for cheap agricultural laborers.

Many of the European immigrants to North America arrived in varying legal conditions: legally bound indentured servants traveling with their masters; semifree "redemptioners" bound to labor for four to seven years (ostensibly to pay off their passage to the colonies); or autonomous but penniless unskilled laborers apprenticed to English tradesmen. Recruiters in England and beyond initially targeted young men in their late teens-early twenties from the middle and lower classes. The early colonial economy was driven in large part by the tobacco plantations along the coast, a labor-intensive crop that created an enormous demand for cheap, semifree laborers. Young men in Northern Europe were recruited by planters, merchants, and speculators, with the promise of a 4-7 years of labor in exchange for passage and freedman status-- which meant citizenship in the colonies or the opportunity to earn citizenship after acquiring land.

In the 1690s, the recruiters began to run out of European men willing to take this deal, and widened their net to more heavily recruit the Irish, Germans, women, convicts, homeless, and orphans. But the population of Northern Europe couldn't sustain this model of immigration, and it became obsolete as planters and merchants increasingly relied on African slave labor.

EDIT: This comment mostly contextualizes the history behind the modern-day issue of illegal immigration coming from Mexico and other Latin American countries. The other commentators have done a great job of answering other aspects about the broad topic of illegal immigration in America.

The concept of illegal immigration in the United States has been present in the country since at least the late 19th century, but it mostly referred to Chinese immigrants at the time, not Latino immigrants. The earliest form of the modern day US Border Patrol was formed in 1904 and mostly targeted Chinese immigrants trying to evade the Chinese Exclusion Act of 1882.

The first notion of a regular armed patrol of the US-Mexico border came during the Border War of the 1910s, when the US Army was deployed to the US Border to protect the US from the fighting going on in the then-ongoing Mexican Revolution. In 1916, the Mexican revolutionary general Pancho Villa was able to organize a raid on the US town of Columbus, New Mexico, killing 98 people and razing the town to the ground. This incident awakened the general US public to the realization that the US-Mexico border was not as secure as previously thought, and led to the formation of a 10,000-strong military operation bent on capturing Villa (that ended unsuccessfully).

With the mobilization of the US into WWI the following year and the release of the Zimmerman Telegraph (which unsuccessfully tried to incite Mexico into a war against the United States), security along the US/Mexico border became greatly beefed up as a result. This change in events greatly angered the Mexican residents of the border town of Nogales, Sonora, who had been used to the freedom of movement/trade between the US and Mexico previously. Tension between the US customs officers of Nogales, Arizona and Nogales, Sonora grew to a fever pitch that finally broke on August 27, 1918, with the Battle of Ambos Nogales, which killed 6 Americans, 120+ Mexicans, and 2 Germans. The aftermath of this battle was the creation of the first border wall dividing the US and Mexico– a chain link fence that divides the two Nogales's to this day.

Throughout the early 20th century, due to a variety of factors (including the social unrest happening in Mexico at the time and the heightened demand for agricultural workers in the American Southwest), millions of Mexicans moved to the US in search of work. Because of the lax immigration laws at the time, not many had the legal documents technically needed to work. Although the Immigration Act of 1917 had began restricting the flow of immigrants into the United States, due to protests from Southwestern farmers who relied on Mexican labor, exceptions were granted to Mexicans.

Nominally, the United States had open borders until the year 1924 with the passing of the Johnson-Reed Act and the Labor Appropriation Act. The Johnson-Reed Act, meant to stem the flow immigrants from Southern and Eastern European countries, also established the notion of immigration quotas from separate countries that still exists today. This act did not affect Latino immigration all that much– under the racial discrimination laws at the time, most Latinos were white. The Labor Appropriation Act of 1924 established the modern day US Border Patrol, though at the time it was meant more to stem the flow of illegal goods into the US than immigrants. In fact, for the first few years of its existence, most of the US Border Patrol's activities were focused on the US-Canada Border than the US-Mexico Border due to the amount of illegal alcohol smuggling into the US from Prohibition.

With the onset of the Great Depression in 1929, the reasons behind these exceptions– the need for more labor– evaporated. With support from President Herbert Hoover, the Mexican Repatriation began in earnest and lasted for seven years, leading to the deportation of hundreds of thousands of Mexicans. Enforced with the help of the US Border Patrol, this event was the first mass deportation event in US history, and can arguably be the first time where the modern day notion of "illegal immigrants" entered the US mindset.