dnd 5e – What happens to an Eldritch Cannon if the Artilerist falls unconscious

You are correct.

As stated in the quote you gave, the cannon only disappears if it runs out of HP or time, or if you actively dismiss it. If you fall unconscious or even die, there is no rule that says the cannon would disappear, so it doesn’t.

The next line answers the rest of your question:

On each of your turns, you can take a bonus action to cause the cannon to activate if you are within 60 feet of it.

Since the cannon only does anything if you use a bonus action to activate it, if you can’t take bonus actions, it just sits there and waits.