You can still buy and bear a cannon with no background check or registration or any of the like, FWIW. Very easy to order on the internet and have shipped straight to your door[].
You can, but be aware that an exploding cannonball (widely available in 1776) is considered a destructive device, so each shell must have an NFA stamp. Solid shot is not considered a destructive device.
[] https://www.dixiegunworks.com/index/page/product/product_id/...