This might be too late for your trip. Hope you had fun.
Putting this here for future travellers.

A lot depends on which bike you ride. I believe Honda, Suzuki, and BMW are all serviced by the car garages from the same brand. Nitro is maybe the largest independent motorcycle dealer (Kawasaki, CF Moto, Beta). Ktm has dealer that sells well, KTM. Also Husqvarna and GasGas. The Dutch version of multibrand motorcycle dealer (like Motoport or Goedhart) does not exist in Iceland.
Riders in Iceland think that motorcycle serviced by a dealer is a cute and novel idea, so there are lot of competent home mechanics with good tools, most with some selection of general spare parts. Asking around near to where you break down is a good idea.
Any part that is in stock in Reykjavík can be almost anywhere in Iceland within 24 hours. If you are willing to pay, almost any part can be shipped overnight to Iceland.

The best Oliebollen are in a supermarket named Fjarðarkaup. Enter the store, cross it diagonally to the opposite corner, where the bakery is. They also sell fresh milk in 1/4 liter containers. Perfect with Oliebollen. Pay on the way out. Enjoy.
Take your time, there is nothing in Iceland that is not worth seeing. Apart from the nature there is a nice motorcycle museum in Akureyri.

PS. Off-road riding is illegal and people are fined if caught. Clearly visible trails are considered roads in that regard. Single tire tracks are not.

Edit. Biking Viking sells and services BMW. Contact Arctic trucks for Yamaha.