My first solid bodied electric guitars and basses had all control cavities and pickup pockets made with drills and chisels, but I was trained by a cabinet maker in using hand tools. Even so it was slow going compared toa router. (dovetail joints, mortice and tenon joints and all that) If you can sharpen a chisel correctly (so that it will cut wood cleanly, with minimum force) and know when to use it which way round, then go for it and work slowly. A very sharp chisel can be a dangerous thing in the wrong hands and and a poorly sharpened or slightly blunt one can be even more dangerous, because if it slips, there will likely be much more force behind it...
These days routers for light work can be picked up fairly cheaply and are quite serviceable for occasional use. I have a copy of a Makita trim router mounted in a sled, for fretboard radius' and it's served me well for a few years. I use higher powered Makita routers for heavier and more frequent work, but most guitar/bass related tasks can be done with lighter powered units, especially if you're not using them all the time. A cheap basic, but serviceable router will likely cost no more than a good chisel and to begin with just buy the bit, or bits, you need, you can always add others later. Learn how to use a router and you'll be surprised at all the jobs you'll be able to use it for.