What about Bombs of various types? Volley guns? Navel mines?
I get that states may have had there own restrictions, so where if anywhere were such weapons were banned?
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.