In reply to tnj: If you're just on the islands to climb, you'll get by fine without a car, just head to Henningsvaer and camp. You'll probably spend half your time waiting out bad weather anyhow. If you're interested in doing some sightseeing, then I'd say a car is pretty necessary - try getting to Utakleiv/Unstad/Eggum by bus.
There's a few local guys around the islands (Svolvaer, Stamsund, Reine) that rent out old cars. Just look for the cars driving around with a 'rent a car' sign on the door and write down the phone #. I've rented from the guy in Stamsund (Odd +47 909 12300) a few times over the years. As of last autumn prices were 500NOK/24hrs, 1500NOK/week (plus an optional 50NOK/day fee if you want to reduce insurance liability).