How much bandwidth does Vassal consume?
Vassal is pretty low bandwidth - as you correctly guess, the graphics are all part of the pre-downloaded game files, so in play it only sends information about which pieces you moved and where to.  It's on the same order as using a text chat app, and pretty favourable compared to a web page.

Obviously you'll need to download and update the actual game files in advance, as those can be larger.

(There's still enough bandwidth usage that you could notice the cost, however, if you're on a pay-per-byte plan for your mobile.)