We ran in to the same problem.....except flying in to Houston and wanting to drive down. There are no car agencies open in Galveston on Sundays.....totally wild in my opinion. If you happened to be in Galveston by Saturday before 5 there is an enterprise there and thats it. They don't open again until Monday.
We ended up renting for the whole week and we will just park it across the street for the 50 dollars it will cost us. It kind of seems like a waste but we would rather have a vehicle and go at our own pace!