One of the highlights for our trip since the very beginning of the planning phase was the ice cave tour, especially for photography. We contacted some other companies to book a private tour to the caves, however they all replied that all the tours were fully booked or private tours were not offered.
All the big companies we contacted always replied after two or three days, which was very frustrating, some others suggested that booking private tours was not a good decision since the only difference in the tour was the private transportation and the cave will always be crowded because of popularity of the place.
As a last shot, once in Iceland we contacted southeasticeland.is to try to book a private tour to the caves, and unlike the other companies, customer service was incredibly reaponsive via email. And being photography our interest, the company offered us a private tour at 4 am a couple of days later after the initial contact, we took and it was amazing.
When we arrived to the cave, no one was there, and the guide made us our life easier by bringing a set of lights and some cool gear like snow axes. It was really nice to be able to see the sunrise on our way to the cave, with an espectacular view to the glacier on an amazing super keep adventure.
Souteasticeland is highly recommended!