Iceland help..

camper rental-- http://www.campervaniceland.com/ https://happycampers.is/ no 1st hand knowledge, but this is but 2 of what a search found --- there were others

hotel-- suggest you try expedia or you favorite search provider for your arrival day-- I found rates from $52- to $900+ in late August

This forum may not be the best source of 1st hand knowledge on Iceland

For most folk here, Iceland likely just a place to change planes, en- route to Europe, not a destination

Returned from a ten day trip around Iceland recently. As above suggest you check Expedia/Trip advisor. My takeaway from visit was that everything was scarily expensive. Iceland doesn't do budget/inexpensive.

The ride from airport into Reykjavik is a bit long, so the hotels by the airport might be a good choice and would make it easy to find a taxi to your pickup point the next day -- they are not fancy but are convenient for just a night. Otherwise, I had great experiences with AirBnB all over Iceland, including Reykjavik. Enjoy Iceland!

When you inquire about the camper van rental, ask the supplier if they have an arrangement with any of the hotels (they often do in other airport arrival cities). Rental agencies usually prefer their renters to have at least one good night's sleep so that they are not driving with jet-lag. If their renters are staying at the same hotel, they might have a "shuttle service" to their location.

We found a hotel in Reykjavik on booking.com that was not too terribly expensive (even for Iceland) for our visit in late August three years ago ... but I tend to book about 6 months out so there is more selection. You could also check on Trivago which compares hotel rates on various booking sites.

Do I need (or is it worth it) to purchase a camping card?
https://www.utilegukortid.is/um-utilegukortid/

I spend a lot of time in Iceland but unfortunately, I don't have any personal experience with camper vans as I always use 4x4 on my travels there. Nevertheless, Happy Campers is the company most people I come across use on their adventures and the one I see the most cruising around the country during colder months but off the F-roads of course.

As to the hotel near KEF airport, I normally use a couple hotels not more than 5-10 minutes away from departures/arrivals. The last one I used, last May, on a 8 room reservation, was Bed and Breakfast Keflavík Airport (http://www.booking.com/Share-8Y4RZx). Strong features are the airport distance, big and clean rooms (recently renovated), breakfast period starting at 4 a.m. and the fact that they provide shuttle transfer to the airport. Be sure to book it during your check-in.

EDIT: By the way. The camping card? I've camped in Iceland for 21 nights in the last 12 months and never used a camping card. If by any chance you need any help with camping sites, please let me know. As a landscape photographer, I know a few :D