Were there U.S. weapon restrictions during 19th century?

I believe this flows from the first phrase concerning "Militia". Militias do did not use mortars, chemical weapons, etc, so the right does not extend to those weapons.

In support of this I found the following on a the site of the commander of a NJ Militia re-enactor:

Cannon were considered the queens of the battlefield. Infantry unsupported by cannon usually lost if the enemy had cannon. American Militia units were known for not standing up against British units with cannon support, since they rarely had any of their own.

I give this example of revolutionary militia because at the time that the Bill of Rights was ratified the militia of the recent revolution would have been in the framers' minds.

Insight into the thoughts and intentions of framers of the Constitution is important in interpreting the Constitution.

There wasn't much of an argument about it at the federal level, and while some states tried to ban/restrict firearm ownership those attempts did not appear to differentiate between what we might consider "military grade" small arms and "hunting weapons" not really suitable for combat. Or small arms in general and larger weapons. I believe there was a fairly powerful motivator for private persons to own "heavy" weapons such as cannon in the 19th and prior centuries which caused this. Namely that of protecting private shipping.

Piracy in the Carribean (though on the downswing) kept going until the 1830s, and the Barbary Pirates resulted in a US expedition in 1801 that lasted until 1805, with another flare-up in 1812-15. Southeast asia also had its fair share. Since merchant shipping needed protecting, it was common to mount cannon aboard merchant vessels. They normally weren't in the same class as warships (3-12 pounders, maybe half a dozen on a bigger merchant ship and tending to be on the smaller side rather than the larger) but enough to make pirates think twice. The thing of it is, even small naval pieces are as big as the field artillery of the day. Since you can't make regulations restricting their use (because then american shipping would be hugely vulnerable to pirates and everybody knows and wants to avoid that) you also can't do much in the way of land-base cannon. So the coastal states and the federal government have a motivator for private ownership of "military grade" heavy weapons.

I know this answer is more in the line of a comment because I don't have sources handy, but I can't seem to comment and thought it might be useful for the asker to know.

I believe that the point of the Second Amendment was that the "common man" would not have enough firepower to fight the "government," but would have enough to form a militia to fight other "incidental" threats. (That is, something like muskets (or rifles) but no cannon, mortars or artilllery.) Such threats could include the following:

1. Slave revolts, such as Nat Turner's uprising or John Brown's attempt to seize Harper's Ferry and arm the slaves.

2. Indian attacks, especially for people on the western frontier.

3. A re-invasion of the United States, or at least American-claimed territory in a future war. This actually happened in 1812, although the militia was not adequate to protect Washington D.C. Arguably this happened in Texas in the 1830s and 1840s if you consider it American "claimed" territory. Also, there was the cry of "54 40' or fight" regarding the Oregon territory.

There have been constant attempts, mostly by cities, to ban guns throughout the entire history of the United States. Nearly all of these thousands of attempts to ban guns, in one form or another, were voided by state court decisions. In some cases states have actually enacted modifications to their own state constitutions expressly making gun possession a right, just for the purpose of shutting down various city and town governments which try incessantly to ban them. The decision by the Georgia State Supreme Court in Nunn v. Georgia (1 Ga. (1 Kel.) 243 (1846)) is exemplary and typical (although in this case it was an actual Georgia state law that was being stricken down):

“The right of the whole people, old and young, men, women and boys, and not militia only, to keep and bear arms of every description, and not such merely as are used by the militia, shall not be infringed, curtailed, or broken in upon, in the smallest degree; and all this for the important end to be attained: the rearing up and qualifying a well-regulated militia, so vitally necessary to the security of a free State. Our opinion is, that any law, State or Federal, is repugnant to the Constitution, and void, which contravenes this right, originally belonging to our forefathers, trampled under foot by Charles I. and his two wicked sons and successors, re-established by the revolution of 1688, conveyed to this land of liberty by the colonists, and finally incorporated conspicuously in our own Magna Carta!” -- Decision of the Georgia Supreme Court 1846

Because of the actions of state courts it was rare for the US Supreme Court to rule on gun ban laws. The most famous such law which reached the high court occurred in the aftermath of the Civil War, when authoritarianism was at a high point in the country. This was the decision Presser v. Illinois, 116 U.S. 252 (1886). Presser affirmed the right of the states to ban people from forming private militias. It left the question of gun control ambiguous and implied that individual states could potentially enact gun control laws. At the time the decision was relatively ineffectual because most states were against gun control.

Overall, the situation in the 19th century was not too much different than it is today: a fight between urbanites and government powers trying to ban guns and farmers and other rural people trying to prevent that. The main difference is that in the 19th century the rural population was much larger, so there was a correspondingly larger failure on the part of gun control advocates.

Did militia companies have cannons? The answer is Yes, they did.

For example, Chapter XLV Militia and Militia Companies, from The History of Detroit and Michigan: Or, The Metropolis Illustrated; a Full Record of Territorial Days in Michigan, and the Annals of Wayne County, Volume 1, by Silas Farmer (1889).

The Legionary Corps, created in Michigan by an act of the Governor and Judges in 1805, included cavalry, artillery, riflemen and light infantry; this was in addition to two regiments of infantry -- all militia.

On p. 317 describes a volunteer artillery company: "On December 27, 1821, on the occasion of the execution of two Indians for murder, the First Regiment of militia was called out and also the volunteer artillery company commanded by Captain Ben Woodworth."

Ben Woodworth kept a hotel, and held several minor posts during his long career in Detroit. The "volunteers" would fire their cannon on the 4th of July every year.