PRELIMINARY FINDING: FALSE
Both of the above statements were made by President Biden in reference to the second amendment. The first claim about not being legally allowed to own cannons during the Revolutionary War was made by Biden during this 2020 Presidential campaign and several historians and mainstream outlets at the time determined it was false. When asked at the time, none of Biden's supporters could point to any laws passed by the continental congress that banned individual Americans from owning cannons. In fact several supporters pointed to laws in England, France, Spain, and even appealled to the laws of China. The problem there is that our constitutional rights as Americans are not dependent on the laws in foreign countries such as England, France, Spain, China, or even Canada or Mexico. When deliberating questions of rights in the US you have to rely exclusively on US history and US legal precedents. This is because at the time of the Revolution there was a complete break between the US and the old world nations not just politically but culturally and economically as well. Since independence our nation has developed its own seperate and distinct legal traditions that don't rely on the legal systems of Europe.
The second statement was made by Biden in the wake of the recent school shooting in Texas. As with the claim about owning cannons during the American revolution, this one is false as well. In fact gun control laws did not exist until the 1934, which is well into the 20th century. Prior to 1934 you were allowed to own both cannons and even machine guns. That means that between the passage of the second amendment in 1791 and the first gun control law in 1934, there were no prohibitions on what types of weapons Americans were allowed to own for the first 143 years of this country's existence.
Because it was contradictory to the second amendment the courts actually struck down 1934's National Firearms Act several times leading to it be amended on multiple occasions to eliminate the unconstitutional provisions.
It is a little known fact that you are actually allowed, as an American individual, to own a tank and even a submarine even today. If you could own those you can certainly own a cannon. You are also allowed to own high explosives. However, if you read the exact text of the second amendment more closely, while you are allowed to own these things, the government has the power to tell you how to store them and how to use them and maintain them. That part is found in the first part of the amendment, "A well regulated militia being necessary...." This means the government can tell you how to store, maintain and even use those explosives or tanks but it can't ban you from owning them outright. It can also regulate when and where you are allowed to carry your guns with you and under what circumstances. But it can't ban you from owning guns for any reason. In some cases, though it can't ban you from owning guns, it can prohibit you from possessing them or having them on your person in certain situations or contexts.
The power of the government to regulate how you handle the arms in your posession was upheld by the US Supreme Court in 1939. However regulating how people handle their arms is not the same as banning them from owning those arms.
With regards to the American Revolution and Constitutional Conventions people did in fact own cannons as individual Americans. They were called Privateers. Those were privately owned ships outfitted with cannons and other privately owned heavy weapons.
Examples of mainstream outlets that have agreed with us that Biden and Democrat claims about the second amendment are false include the Austin American Statesmen and the Washington Post.
In fact as the Post noted, the right to bear arms has only been a right reserved for US citizens. Native Americans and Blacks were not allowed to own arms because at the time of the Revolution and the ratification of the US Constitution they were not US citizens. Upon becoming US citizens in the 19th and 20th centuries they acquired the same rights to own arms that all other US citizens have.
As the Washington Post points out, Article 1 Section 8 of the US Constitution allows the US government to contract privateers and even private militias to help fight America's wars.
We find that Biden is just as bad at American history and the US Constitution as Donald Trump ever was.
Final Finding: False.
The government has no power to ban you from owning cannons but as with other stuff like submarines and machine guns the government can regulate and tell you how, when, where, and in what circumstances you can have them in your possession, how and where you can store them, how to maintain, and under what circumstances and in what manner you can use them. As an example they could ban you from having guns in your possession if it was found you were mentally unstable but even then it could not outright ban you from owning them. But as we noted regulating how you maintain something is not the same as outright banning you from owning that something. Hence we rule that Biden's claims and by extension the Democratic Party's claims about the second amendment are 100% false.
Other media outlets that have also found the claim to be false: