Can I do a raid with a full Pokemon Box?

You can do the raid with the full Pokémon box. You cannot however capture the raid boss afterwards directly; you'll get a message saying your box is full (and if you haven't upgraded to 1000 yet, to upgrade in the shop). Then it will kick you out to the map (unless you click "Go to upgrades" in which case it kicks you to the shop, but either way the raid capture is done for now).

If this happens, you may still be able to capture the boss. You'll have to first transfer a Pokémon or upgrade your storage. You of course can instead ignore the raid boss if you prefer.

Then, assuming the boss hasn't left the gym yet (i.e., the timer hasn't expired), and assuming you're still "in range" of the gym, you can tap on the gym, and it should send you directly into the capture interaction. This does seem to have some bugginess in it; you may or may not be able to know the CP you're getting (there were some bugs there with re-entering showing you the wrong CP - see issue 12 here) and sometimes you get "Error" and can't do it. In my experience I've never had an issue re-entering, but some people have issues regularly.

Note that this only affects the Pokémon storage; the raid items will always be delivered, letting you go well over the maximum item storage.

According to one Reddit, you are forced to transfer a Pokemon from your box before you can capture the Raid pokemon, or you can upgrade your space via purchasing upgrades. You will be exited from the gym, and will have to enter it again to fight the pokemon (as stated in the comments and other answer).

They include a gallery of pictures for proof. So yes, you can do the raid, but will have to transfer a pokemon (or buy more space to hold it).