Can I (a 13 year old) buy supplies for a cat by myself with no parent there? Or do I have to bring my older sister (an 18 year old) with me to get them? Or do my parents have to go with me?
As long as your money is green and you aren't buying something prohibited to youngsters (booze, cigarettes, etc.) I doubt any store is going to refuse your business.
My 12 year old routinely makes purchases with cash or a gift card (either a store's card or a Visa/Amex card that acts like credit card but is a gift card) and has never had an issue. Clothing, make-up, bath items, etc. I understand in some areas you need to be over 18 to buy certain markers, spraypaint, or other propellant items that can be fatal if inhaled. I see little issue with buying pet supplies, but it wouldn't hurt to have your sibling nearby if you think there will be an issue.
I had a cat growing up--most of the time I was the one who got her supplies. It was never an issue.
Perhaps a technicality, but minors do not have the legal capacity to bind a contract. Making a purchase from a store is a contract. I'm not a lawyer and there may be case law to the contrary or that creates exceptions, but my understanding is that purchases made by a minor may be void if later challenged.
JohnFx's answer is true from a practical sense. But if you get turned away at a store, understand that they're probably just being careful to avoid headaches later.