Could inbreeding cause allergy issues in a cat?

I rescued my cat from a farm when she was a kitten. I grew up on a farm so I know how short a farm cat's lifespan can be.. I wouldn't be surprised if her parents were closely related. She's a smart, normal kitty, but she has really bad skin allergies. She'll get severe scabs all over her chin and chest. At first we were told it was cat acne, but our newer vet said it's allergies and she requires a yearly shot. Is it possible that things like allergies can be more prevalent in cats who are inbred?