I know how to buy Cutlasses from the armory, but how do I load them onto the ship? Also, what are the requirements for buying cannon and, likewise, loading them onto my ship? I want to get a convoy started, dernit.

You can transfer cutlasses (and cannons, for that matter) by selecting the ship, clicking on one of the cranes and selecting "Weapons" from the menu.
There are no requirements for buying cannons (or at least none that I know of), but your ship must be capable of carrying them. Cogs/Hulks and upgraded Crayers can be equipped with cannons.

Marvelous, thank you. As for cannons, I think my problem is that I simply can't purchase them in the first place, them or bombards. How do I find a city which crafts and sells these?

I've checked, and in some randomly selected towns, I could buy cannons and bombards with no problem. However, it's 1365 already in that game, so towns are rather advanced.
Maybe the weaponsmith is currently out of those?
Or, it could be that, like with Hulks and shipyards, towns have to reach a certain point of development to produce the most advanced ship weapons.
Of course, these are only my guesses; unfortunately, the manual doesn't say anything about their availability; that's why I think maybe you just have to wait for the smith to restock.
Anyway, you can use less powerful armaments (ballistas and catapults) for your convoy.

The weaponsmith in a town has the same technology level as the shipyard, so the more advanced the ships you can build, the more advanced also the weapons you can buy.

Keep the town well supplied in Hemp Timber Iron goods and Leather as well. He can't produce if he can't buy the raw materials. Look at the shipyard, there are 1-3 "lanterns" on the dock. When you get 2-3, you should start seeing bombards and cannons available.