Article 1, Section 8, Clause 11 of the United States Constitution allows for Congress to grant Letters of Marque and Reprisal, which were essentially permits to allow private citizens to capture foreign vessels during wartime. Essentially, government allowed piracy. It is therefore assumed that those private citizens would own ships outfitted with cannon and therefore were able to own cannons.

But, that doesn't exactly answer your question.

The BATF in the United States regulates firearms and firearm ownership. s 18 U.S.C. § 921(a)(3) defines a firearm as A. any weapon (including a starter gun) which will or is designed to or may readily be converted to expel a projectile by the action of an explosive; B. the frame or receiver of any such weapon; C. any firearm muffler or firearm silencer; or D. any destructive device. Such term does not include an antique firearm. [emphasis mine]

The term “antique firearm” means any firearm (including any firearm with a matchlock, flintlock, percussion cap, or similar type of ignition system) manufactured in or before 1898. The definition includes any replica of an antique firearm if it is not designed or redesigned for using rimfire or conventional centerfire fixed ammunition, or uses rimfire or conventional centerfire ammunition which is no longer manufactured in the United States, and which is not readily available in ordinary channels of commercial trade. Further, any muzzle loading rifle, shotgun, or pistol which is designed to use black powder or black powder substitute, and which cannot use fixed ammunition, is an “antique firearm” unless it (1) incorporates a firearm frame or receiver; (2) is a firearm which is converted into a muzzle loading weapon; or (3) is a muzzle loading weapon which can be readily converted to fire fixed ammunition by replacing the barrel, bolt, breechblock, or any combination thereof. 18 U.S.C. § 921(a)(3), (a)(16), via https://www.atf.gov/resource-center/docs/0501-firearms-top-10-qaspdf/download

Federally, cannons are not regulated even today.
26 U.S.C. 5845; 27 CFR 479.11 via https://www.atf.gov/firearms/qa/are-muzzleloading-cannons-considered-destructive-devices

From an 1899 book "A History of American Privateers" by Edgar Stanton Maclay "The Constance-Warwick was the first frigate built in England. She was built in 1649 by Mr. Peter Pett for a privateer..." "She measured four hundred tons and carried twenty-six guns, divided as follows: Eighteen light demi-culverins, or short 10-pounders, on the main deck; six light sakers, or short 5-pounders, and two minions."

Later in that same book: "By an act made June 26, 1812 [quite a few years after 1791], the prize money from captures made by private armed craft was to go...." This would show that in 1812, 21 years after the ratification of the Bill of Rights, there was still privately owned warships, and we have previously established that those warships carried heavy guns.

This book is fascinating reading. It is too much to continue to quote, but there are NUMEROUS examples of private ownership of cannons in it. Lots of other fun reading too, but also cannons.

Town militia companies often formed as artillery batteries, but I doubt many cannons were owned by individuals within a community. While militia requirements often specified the exact type of musket or other sidearm required for the specific service they were expected to render, something like a cannon - and its limber and caisson and other necessary elements - would have been owned by the community itself. That is, it would have been owned and operated by a somewhat nebulously defined semi-legal entity like "the citizens of Cannontown" rather than any particular person in the town. The men of the Cannontown militia, then, would be responsible for the storage and upkeep not only of the cannon(s) but also the ammunition and necessary equipment for their use. This would likely take the form mostly of regular dues or payments to cover costs. This meant, also, the erection, staffing, and maintenance of a "powder house" or magazine; poorly kept powder was dangerous for everyone. It should also be said that most towns in the early United States would have had powder stores where most of the men of the militia may have kept their arms and ammunition so it wasn't in their house, but they could access it at need. Again, black powder lying about unstored was highly dangerous.

Acquiring the artillery would also likely be somewhat more of an undertaking than just making an order and picking it up from the local foundry. Production of artillery was expensive and involved specialist equipment and expertise, and many town militias with access to artillery may have had "grandfathered" pieces, either taken, purchased, or produced for earlier conflicts, and then given over to artillery companies of the militia. The exact form of these transactions is something I've never studied or come across, unfortunately. It's possible that individual towns or artillery companies could order privately from ironworks and other foundries, or simply use the state or federal government as an intermediary, I'll have to look and see what I can find out about the specific dynamics of this.

It should also be said that some of the main targets of rebellions or acts of violent political resistance were powder houses and armories which housed artillery pieces. Shaysite regulators marched on the Springfield armory, and Whiskey rebels often copied the revolutionary war tactic of removing locally-owned powder and weapons from communal stores before trouble started.

So, if the question here is about whether the 2nd Amendment covers the access of American citizens to pieces of artillery, the answer is undeniably yes. However, it did that in a social context in which weapon ownership and bearing was intensely communal, with structural oversight by, at least, local peers, if not the state and federal government.

This statement by Joe Biden has been fact checked and found to be incorrect.

In 1791 there were no federal or state laws limiting the purchase or ownership of cannons. The first state laws which regulate weapons were passed by states in the early 19th century, limiting the possession of concealed weapons (beginning with Kentucky in 1813). Cannons were not restricted by either federal or state statutes and were in fact owned and used by privateers. That does not tell us how Congress or the legislatures of the several states would have reacted to private acquisition of cannons that seemed more along the lines of someone raising a private land army. Whether you could acquire cannons in such a manner and whether Congress or the States might have responded to such acquisition with new regulations is hard to impossible to say because, as far as I can determine, no one ever tried. Making the question even more difficult is the fact that the legal doctrine needed to form an answer did not exist in 1791. (See re: Marbury v. Madison, 1803, below).

There is daylight between the absence of laws restricting gun ownership or possession and the Second Amendment positively protecting those rights against hypothetical state laws restricting or regulating that right. That is, just because states didn't pass laws outlawing the private ownership of cannons didn't mean that they couldn't, or even that they wouldn't do so had private individuals started acquiring them in great number. To the extent that Joe Biden was trying to say the Second Amendment did not protect the right to own a cannon, or, more generally, that the Second Amendment was not intended to create a private right to own any weapon one desires, he might be right.

Whether or not the federal government could have limited the ownership and possession of cannons notwithstanding the Second Amendment (perhaps pursuant to their exclusive right to raise armies to and enact laws that are necessary and proper under Article II Section 8) is debatable. Marbury vs Madison was not decided until 1803, so even the ability of the Supreme Court to strike down laws they found unconstitutional was not yet established. In fact, the famous holding of Marbury vs Madison was decided in dictum -- that is, they decided the Court has the ability to strike down unconstitutional laws without actually strike down a statute of substance. This may even have been an intentional tactic to make the holding of Marbury more palatable for legislators and other members of government. Creating the rule of Marbury while striking down a state law, especially a popular one, would have been more controversial. It is impossible to know how the states and the people would have responded, or whether the authority of the Court would have been respected.

Whether the states could have passed laws limiting the possession of ownership of cannons is less controversial. With the possible exceptions of states that protected the right to have and bear arms in their own state constitutions, they almost certainly could have. This is due to the absence of the Fourteenth Amendment, incorporated after the Civil War in 1868. The Fourteenth Amendment and the doctrine of Selective Incorporation is crucial to understanding constitutional law in the modern era.

Prior to the passage of the Fourteenth Amendment, the federal constitution did not directly limit the powers of the states in any manner prescribed as part of the Bill of Rights (the first ten Amendments of the United States Constitution). The Second Amendment, along with the rest of the Bill of Rights, was a limit on what the federal government could do. States were restricted primarily by their own constitutions, only a few of which provided protection for the right to bear arms. Pennsylvania's Constitution of 1776, for example, provided "That the people have a right to bear arms for the defence of themselves and the state; and as standing armies in the time of peace are dangerous to liberty, they ought not to be kept up; And that the military should be kept under strict subordination to, and governed by, the civil power." Whether a Pennsylvania law violated that provision of the Pennsylvania Constitution would be a question for the Pennsylvania Supreme Court. There would be no federal issue for the United States Supreme Court to review, and so the case would end there with whatever ruling the Pennsylvania Supreme Court provided.

Since the passage of the 14th Amendment, Amendments from the Bill of Rights have been incorporated one at a time, meaning they now apply to the states. In 2010 the Second Amendment was the last Amendment to become incorporated in this manner with a case called McDonald v. the City of Chicago. This case means that now the states' power to regulate the right to have and bear arms is equal to the power of the federal government to do so.

Edit:

The simple answer is that no restrictions existed in 1791. I hope the context I provided, including constitutional and state constitutional framework underlying whether and how such restrictions might have been promulgated, and how Second Amendment doctrine worked then and how it changed over time, is helpful. I've never attempted to answer a question on this sub before and my background is as a lawyer and student of constitutional law rather than historian, so if anyone has constructive criticism about how to more completely or appropriately form an answer I would appreciate it.