I'm afraid I don't know of anyplace you can get an antibiotic without a prescription -- thank God for that, because misdiagnosis and incorrect use of antibiotics can cause more serious problems than the one you're talking about. First of all, you didn't really give a description of your cat's symptoms, so I don't know if there's even an infection or if you're just assuming it's infected. Even if it is infected, you should know that there are different types of infections (for instance, streptococcus, staphylococcus, etc.) and the appropriate antibiotic can only be determined once the type of infection is determined, since each antibiotic will only work on certain infections. Also, the correct dosage has to be determined based on your cat's weight and the type of bacteria that needs to be killed. Unless you're a vet, you can't make these determinations on your own.
Considering the fact that infections NEVER go away on their own (bacteria continue to multiply as long as they live), the infection will only get worse if it's not appropriately treated. And the only way to get it appropriately treated is by taking your cat to the vet. So PLEASE do that right away!