Joints in wood move.
When we build with wood we are using a material that is inherently lacking in dimensional stability. Wood continually takes in or gives up moisture depending on temperature and humidity of the air and the piece of wood. As it does this it expands and contracts in size. Air dried wood is less stable than kiln dried, and it varies with different species , as well as with, or across, the grain and finish on the completed piece. Plywood is more dimensionally stable than boardwood, and manufactured wood like chipboard,particle board, and masonite even more so.If you go to an antique store the proprietor will tell you that the loose legs, split tops, end grain cracks, etc. are part of the charms of a handcrafted item with natural materials, so you should buy this old piece of junk that's falling apart: because flaws are charming.
Well, perhaps "flaws" are not so charming when they're from errors, but the marks that something was handcrafted and has attained a patina of use are. Make your wagon to the best of your abilities, but don't let "the best become the enemy of the good." Don't avoid making a wagon (or any other project) just because you can't produce the equivalent of a "royal coach."
I think it was Tage Frid who said something to the effect that the accuracy required for joinery in fine woodworking exceeds the accuracy obtainable by our measuring and marking devices. This impacts our wagon building in several ways.
We obtain a tight and true joint by trial fitting. We cut close, then trim, file and sand closer. The beginning woodworker should cut further from the line than the experienced woodworker. Take your time.Don't expect that you will cut pieces to the exact dimensions on a plan, and poof! it magically fits perfectly. Wood isn't like that. Be aware that if you make a nice tight egg-crate joint (for example) unless you keep your wagon in a temperature and humidity controlled environment forever, sometimes the joint is going to be too tight, and sometimes it's not. I always bring a "persuader" (sometimes called a mallet) with me to events for when they're tight, and you can shim if you need to when they're loose.
If you find that you've cut the joint too loose to start with don't worry, there are a bunch of fixes. For just a slight over-width take a piece of heavy paper wet it with glue and apply it to the side of the joint. In effect you're papier-macheing in a permanent shim. For a little bit wider error you can tack and/or epoxy in a shim of sheet metal, for an even wider joint: leather. And if it's way off the reservation cut a wood patch and glue it in place.
As I've noted elsewhere, I like wagons that breakdown for transport, but you can glue/screw the joints if you are willing to sacrifice portability for security.
Some other tips:
-- use the same ruler for all measurements and cuts, they're all just a tiny bit off from each other.
-- a "story stick" is a piece of wood that you mark of one measurement on to transfer it to another piece. In many cases the actual units of measure don't matter nearly as much as the relationship of the parts.
-- you can't cut a long perfectly straight line by hand, use jigs or fences to guide your saw
-- dull tools screw-up your work. Don't economize with old, worn, blades or bits.
Most of all, when you roll up your new wagon to your friends at an event they are going to be hugely impressed and complimentary, trust me, you'll see a conglomeration of every flaw-- they'll see a wagon.
You really can do it.