Did the 2nd Amendment really limit the purchase of cannons from the day it was passed?
No. The 2nd Amendment protects "the right of the people to keep and bear arms" and does not single out cannons or any other weapon individually.
The Second Amendment states simply (.gov site)
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms shall not be infringed.
and does not mention anything about cannons specifically (e.g. that the people do not have the right to keep cannons).
PolitiFact looked into Biden's statement and concluded it was false. They mentioned documented examples of privateers, who are individuals, owning cannons during the American Revolutionary War.
Here's a relevant snippet:
The Biden campaign didn’t offer evidence to support his statement,
and historians of the period are dubious laws against private cannon
ownership existed.
There are documented instances of privateers, or privately owned
vessels that legally attacked enemy ships during wartime, setting sail
with cannons during the period.
[...]
But the Biden campaign was unable to point to a specific law. "The vice president's point is that to help end the tragic epidemic of mass shootings that is taking so many American lives, we need to ban weapons of war from our streets," the campaign told PolitiFact.
Historians say they are doubtful that there were laws to bar individual ownership of cannons during the Revolutionary War period.
"It seems highly unlikely that there were restrictions on the private ownership" of cannons, said Julie Anne Sweet, a historian and director of military studies at Baylor University.
[...]
The 1899 book "A History of American Privateers" by Edgar Stanton Maclay notes several cases in which privateers during the Revolutionary War set sail using cannons.
This is my guess as to how the claim started appearing. The claim has been circulating for at least 8 years (without a connection to Biden).
In the 1980 case State v. Kessler, this is what the Oregon Supreme Court wrote of the term "arms" when trying to interpret the intention of the writers of the Oregon Constitution (and, indirectly, the Indiana Constitution):
The term "arms" was not limited to firearms, but included several handcarried weapons commonly used for defense. The term "arms" would not have included cannon or other heavy ordnance not kept by militiamen or private citizens. (from Justia)
It was apparently a point of discussion whether states could restrict gun rights more than the US federal government as the 2nd Amendment prevents the federal government from infringing on this right, but does not necessarily prevent the state from doing so as well (explicitly at least). This essay on congress.gov goes into detail on the issue. In the present day (only recently in 2010), the Second Amendment has been determined to apply to states.
And, more importantly, because the District of Columbia is a federal enclave, the Court did not have occasion to address whether it would reconsider its prior decisions that the Second Amendment does not apply to the states. The latter issue was addressed in McDonald v. Chicago [a 2010 Supreme Court case], where a plurality of the Court, overturning prior precedent, found that the Second Amendment is incorporated through the Fourteenth Amendment and is thus enforceable against the states.
The first sentence of the essay sums up the source of the confusion well.
For over 200 years, despite extensive debate and much legislative action with respect to regulation of the purchase, possession, and transportation of firearms, as well as proposals to substantially curtail ownership of firearms, there was no definitive resolution by the courts of just what right the Second Amendment protects.
