Is it possible to make onion Bhajis without a deep fryer? I love these things. Is it possible to cook them in e.g. a frying pan if I keep turning them around? Or maybe by just filling up my crockpot a little way with oil (enough to cover the Bhajis but not so much it will spill over or spit everywhere)? Is that safe?
Yes, you can cook them in about 1" of oil in a steep sided frying pan or wok. Just turn them reasonably frequently to ensure even cooking.

If you're dubious about using a lot of hot oil, you can use less by making your bhajis a flat, patty shape and shallow frying.