A contract is more than a bureauocratic hoop to jump through. In the case of dispute or even misunderstanding, it's the document that keeps both you and your contractor safe. Say your contractor lays linoleum where you wanted hardwood flooring, or lays insulation bats where you wanted it blown in. You agreed on a price for very specific work, and the contract should spell that all out. (Likewise, if you agreed on linoleum and are now demanding hardwood, it's the contract that protects your contractor.)
A contractor who doesn't want to sign a contract isn't one you want. In the same vein, a contractor who does good work should be able to get permits for reasonable work. And a contractor who tries to talk you out of the need for these permits is suspect. Is she lazy or is she trying to cut corners? Either way, you don't want any part in it.