Many folks, me included, have rebuilt or just built cabinets with birch ply. It takes stain very nicely and is light and strong. There are two varieties, my choice is American birch sold in 4x8 sheets at most home improvement type stores. Also, there is true Baltic birch sold at specialty wood stores, comes in 5x5' sheets. Both have fewer voids than regular plywood and is a great choice for cabinets. I use 3/4" inch and T-mold the edges of almost all of the cabinets I've built. The T-molding is available from video game suppliers (Ebay or online) and a router bit to groove the edges of the ply is available about anywhere. I have a few builds on here with extensive pics and examples. Glad to get more specific if you like. The home-build section here has lots of rebuilt Sportsmobiles and one-off clones, etc.
If you like the layout of your van you could easily use your existing cabinets for templates of new ones out of different, better wood. I have done this on a few vans. Sportsmobile's cabinets are just held together with L-brackets and can be cleanly taken apart into a nice, flat stack of wood. In other words, nothing is destroyed if you take yours apart to use as templates for new. It's pretty simple if you have access to a circular saw, jigsaw and/or table saw.
Here's an example of a modified Sportsmobile with lots of birch ply and a simple polyurethane finish:
Linear Mode
