Dogs in cars

I usually take my dog EVERYWHERE with me. 
She is not a service dog so I don't know if they'll allow her with me at stores. Although on hot days I always sneak her in with me because I'd rather her come then to suffocate in all that heat. 
But on normal days I leave my windows open enough so she can breath. Is it possible I get a ticket for doing this or should I always sneak her in stores with me. She's a little pup. She fits in a purse.