We did that tour 3 years ago. Booked in March and the company we used (tour.is) was only able to give us one starting date in August as other dates were booked up. Info and customer service they provided to us was perfect.

Self driving in Iceland is easy. Roads are well marked and there aren't enough of them that you could get lost.

For this summer you should have reserved back in December or January.

Maybe there have been cancellations and you can still get in now?