Yes, most rental cars have manual transmissions. If you need an automatic you will have to reserve it well in advance. Make sure you re-confirm this reservation just before your arrival and make it with a reliable company. On Santorini I always recommend Vazeos in Oia, Tony's in Firostefani and Spiridakos in Fira. Spiridakos also has an office at the ferry port if you're arriving by sea. They all have websites.
A rental car on Santorini is a good idea. It gives you the most flexibility for exploring the island. The bad news is that parking, although free, can be difficult to find in August because of the volume of cars on the road. With that in mind try to find the smallest car possible with an automatic transmission.
If you live outside the EU be sure to bring an International Driving Permit. If you live in No. America you can get one at your local AAA/CAA office, no test or membership required.