Short answer: Yes you can. Also, no need for VPNs etc. You can simply go to any of the various country-specific websites (budget de, budget.co.uk, budget.ch....) to shop around for prices without having to switch residency around at all.
Pretty much the only difference I have found is that the number of miles included in the price varies. The German site generally offers a 200 mile/day allowance while the UK site includes 300 miles a day in its (higher) price.
Booking and paying through an European site also gives you better protection in case of things going wrong as EU consumer protection law applies.