The short answer is yes, and that naturalization from Great Britain was even encouraged. In fact, some of the policies and decisions made by the US government in the lead up to the War of 1812 were about protecting the right of naturalized American citizens of British extraction to live and work as Americans.

But obviously it's pretty complicated. The United States was in an odd place following the War for Independence. The government hadn't stabilized much, and domestic disputes and violent rebellions continued throughout the 1780s and didn't stop after the ratification of the Constitution. In fact, "establishing uniform Rule of Naturalization" was one of the structural goals of the Constitution itself. The reasons are complicated, but in essence the British interfering with a colony's control of its naturalization policies was one of the grievances that led to the War for Independence.

James Madison, in Federalist 42 argued that states overseeing their own naturalization, though, was a fault of the Articles of Confederation. He argued that "America was indebted to emigration for her settlement & prosperity" and that the United States should make an effort to "invite foreigners of merit & republican principles" to emigrate. This was a generally popular and relatively unopposed sentiment, but the thing is that the framework of "merit and republican principles" was somewhat self-selecting, and it mostly meant British people. This can be seen in a couple of controversies in Adams' administration.

Following the French Revolution, the American relationship with France was ambiguous. While elements of the American political elite had always been friendly to France, the scope and violence of the revolution and its ability to sustain a stable government were by no means clear. Tensions tightened further when French ships began taking American prizes bound for trade with England (as a subscript here, you'll note that American trade with Britain resumed almost immediately following the War for Independence and continued almost uninterrupted for some time... we'll get to some complexities later). Adams sent a commission to France to negotiate and clarify the relationship between the two countries. Controversy was almost immediate, as the French ambassador Charles Maurice de Talleyrand approached the American commission through agents (named X, Y, and Z in letters eventually published in the United States), demanding a bribe before formal negotiations could begin. This was evidently a known practice, but the American commission was offended, and publicly refused. Charles Pinckney, one of the commission, famously blurted "No, not a sixpence!" when the question was raised.

A combination of Revolutionary France's bellicosity, the formalization of what came to be known as the First Coalition and its resultant war, and the scandal of the "XYZ Affair" led John Adams to declare a limited naval war against France called the Quasi-War, but it had knock-on effects of the Federalist party in the United States taking formal efforts to control and censure American politicians with relationships to the French, eventually leading to the Alien and Sedition Acts, which allowed the president the power to imprison or deport non-citizens for nebulous reasons, made it more difficult to become a citizen, and punished criticism of the United States government. Adams was extremely reluctant to sign it, believing it an unbalanced expression of executive power, but he did, to great public outcry.

But you'll note this was specifically about enemy countries and in this case it meant The French, especially in the wake of the astonishingly violent French Revolution. Naturalization with British expatriates continued uninterrupted, and was especially prevalent in seaport towns, as the decades following the War for Independence saw the United States build up to the largest merchant fleet on the Atlantic, and all those trade ships were hungry for experienced crewmen. Because of pressures of a decades-long war against Revolutionary France and then Napoleon, the British were engaging in a practice called impressment, or drafting British citizens into the Royal Navy wherever they found them. The issue became one of sovereignty to the United States: two years of residence was enough to naturalize a British citizen to an American one, but the British didn't see it that way, and furthermore claimed jurisdiction over Royal Navy deserters, regardless of their latterday political loyalties. This was a huge issue in part because the British regularly stopped and searched American vessels for British citizens and especially deserters from the navy.

In combination with other factors, this led to agitation against Great Britain, helmed chiefly by Thomas Jefferson. Taking inspiration from 'the troubles' that led to the War for Independence, Jefferson decided in 1807, following the Chesapeake Affair - about which much more here - to pass restrictive act on American trade. The idea was to strangle trade with Great Britain so they would recognize the value in American trade, but Jefferson essentially made foreign trade itself illegal in the Embargo Act, and many saw it (rightly) as a partisan attack against his political enemies, who were predominantly coastal merchants or east coast elites whose livelihood was tied in with foreign trade. It was deeply unpopular and economically ruinous, and two years later it was replaced by the Non-Intercourse Act, which only made trade with France or Britain illegal.

However, smuggling and illegal trade flourished under these acts, and smuggling was especially lucrative along the Canadian border. During this time immigration itself wasn't at issue - American ships continued to hire foreign hands when they could (and when they could legally or illegally sail) and British hands were especially critical, as they spoke the language and were experienced at sea. The British, of course, did not slow or restrain the pace of their impressment, and continued targeting American ships.

This of course was one of the chief tensions in the War of 1812 - on which I elaborate here - which Madison (now president) sought to solve with the invasion and seizure of Canada. One of the reasons that course of action seemed so easy - Jefferson infamously quipped that it was a mere matter of marching - was because a great deal of territory in Upper Canada (modern day Ontario, mostly) was peopled by American emigrants. The thought was that these people were obviously loyal to the United States, or would at least be friendly to American troops.

The reality was much more difficult and the United States never meaningfully succeeded in any of their invasions, and their invasion in fact helped to create the basis for Canadian national spirit, though it's hard to quantify something like that, obviously.

Long story short: immigration was connected to many political issues of the first decades of American national sovereignty, but immigration bans or suspicion was never the core issue, and apart from the years between 1807 and 1815 or so, it would have been completely routine for UK citizens and ships to call and trade at American ports, and vice versa.