In reply to oceannash909:
Sadly I've not been to Lofoten for quite some time, but I have done I think 3 week long trips there since the 90s.
It looks like there are now some basic facilities that you pay a bit for at Kalle Beach https://goo.gl/maps/qVDn2gvWExGnz7M58 but the google reviews suggest there are also limitations on how long you can stay there and so on.
The other traditional place climbers camped was below Gandalfveggen - approximately here_ https://goo.gl/maps/3xeV1Cgzp8MKoD6W6 and from street view it looks like it is well used by camper vans. But even in the Noughties I remember there were problems with too many people camping there and, to be frank, leaving piles of crap and toilet paper not very well hidden.
To be honest - I think you are going to find it hard to get the best out of the islands without a car, but good luck and remember the weather is always perfect up there (or at least it always has been when I've been there). My top tip is take crack gloves or make reusable tape ones, as you climb much faster with them and climbing mainly cracks for a week is pretty hard going on your hands.