The 'average Joe', particularly if living in the countryside, rather than in a city (a very small % of the US population lived in cities in 1791) would quite likely have owned a musket, if he lived in the East. Possibly a rifle, if he lived nearer to the frontier in the West.

Muskets were more useful for hunting birds (as they could shoot birdshot like a shotgun) like waterfowl and turkeys, (which were more often hunted in the East), but they could still be used (loaded with ball or buckshot) to hunt deer or bear, and for defense or militia duties. In the west, where larger game was more prevalent, the accuracy and range of a rifle, made it a popular weapon.

Your 'average Joe' would have had little use for more advanced weaponry (and might not have been able to afford it), but private citizens certainly could and did own cannon.

Privately owned merchant ships often mounted cannon, and a few years later, in the War of 1812, privately owned privateers certainly did.

This site lists over 500 American privateers operating in the War of 1812.

http://www.1812privateers.org/United%20States/index.html

Privateers were privately owned ships, operating under a "Letter of Marque" from the US government, which allowed them to war against enemy vessels.

As these privateers were armed with cannon, and were privately owned, we can safely say that private ownership of cannon was certainly legal in the USA after the Constitution was ratified.