Typical Fender-style flatwork is vulcanized fibreboard- https://en.wikipedia.org/wiki/Vulcanized_fibre- which is an excellent material for the job for a variety of reasons.
That being said, you could use a variety of other options. Pretty much any reasonably stiff non-conductive material could be used: most types of plastic, circuit board materials like phenolic or glass-epoxy composites, formica, etc. Wood could be used, but I'd be concerned about movement with humidity changes putting stress on the windings. Also, in a typical single-coil where the magnets are a press-fit, I'd worry about getting a hole drilled cleanly and accurately enough in wood to provide the correct fit. And again, the movement with humidity could easily see the wood cracking, or the magnets falling out. Essentially then, solid wood looks like a bad choice. No reason you couldn't use something like very thin birch plywood, but it would cost more than the fibreboard.
I'm not a winder myself, but the fibreboard seems almost ideal. It is easy to work with, not brittle, inexpensive, and available in various thicknesses and colours. If you don't keep it in a damp basement it won't rot, especially if the pickups are potted in wax or similar.