Power tools didnt always exist. You can sure as hell do it without a router, but it requires MUCH more patience, MUCH more time, and MUCH more care. Basically, get yer chisel out, and hack
More realistically, (i havent tested this but it seems fairly logical) I would say you could try to drill a bunch of holes to the approximated depth of each cavity you need to cut (say 50 holes + for 1 pickup cavity) so you are removing most of the wood. Then get a chisel out and scrape the extra, a few multiple sized chisels might help.
With this method I'd say you're probably looking at close to 3-4 hours per cavity, in which case, who the hell wants to do that?
Go to Sears, or look up Harbor Freight online. They sell good/decent plunge routers in the range of $50-$70 that will do any job you need to do, and with far less effort. You'll also have to pick up a few bits (template cutting bit, flush trim bit, maybe even a straight cutter bit) which could cost you about $25 more. Even then you're lookin at less than $100.
If you plan to do more than 1 guitar, don't kid yourself, go get a router.