Do gas stations run background checks?

Yes, they may. You are dealing with merchandise, selling gas, and selling alcohol (in some states), and selling tobacco. They want to make sure that the people they are hiring will not rob them blind because gas stations only make 3 cents on every gallon of gas and really they are penny profit businesses so they will definitely run a background check, especially if you are an adult, to see if you are who you claim you are.