​Here are the best car rental companies in 2018. Some need a credit card and some don't. I will make a list for you and keep you informed on any changes in the future.

Hertz,Fox, Alamo, and Avis currently accept debit cards. There may be a deposit or a credit check to use their service.

Some of the big names offer debit card rentals in other countries.
I have worked with all three of these companies and recommend them, not just for great prices, but excellent customer service.