If you are going with dead axles, why don't you just support the axle/bolt from both sides with two chassis rails on each side of the robot chassis? This would take a lot of the stress off the single hole and spread it out between two holes.
Drilling precise (though not necessarily accurate) holes can be easily acheived by clamping frame members together (possible putting some screws through to guarantee they won't slide), and drill all of them at once on a drill press.
The outer chassis rail can either be permanently attached, and have the axles able to side out of the robot frame, or you can press threaded wood inserts into the wood to have a removable outer chassis rail with bolts.
As for the existing design, the only frame members that you will really have to worry about are the front and back lateral members in high speed impacts, which may splinter the wood. To help protect against this, use a biscuit joiner and drill biscuit holes in the tops and bottoms of these laterial members, and attach a 3/8"-1/2" by 1.5-2" wide piece of wood across the top and bottom to make a C-channel. If you use a lot of biscuits and wood glue, this should form a joint stronger than the original wood itself.
Actually, the more I think about it, any team which may lack a machine shop or CNC sponsor can create a really impressive robot as long as they have at least one parent/mentor with a bunch of woodworking tools and know-how.
Onto this being a "cheap" way out, I wouldn't really think of it as such. Perhaps we've gotten so used to aluminum and polycarbonate in FRC that we forget that wood can have a lot of interesting material properties that make it ideal in some applications. And if done right with careful attention to detail, using a router to make all the edges nice, remembering how to position the wood grain to maximize strength, and staining all the wood with a nice, rich color, then the final result can really be quite impressive. In fact, if a team went this far, they might even just carry on the theme and make everything
steampunk.