Can you get money back if you didn't sign the contract?

This is a vague question and more details would be helpful. It sounds like you made a payment while aware of the terms of the deal. To me, it sounds like you will be held to those terms, since you knew what they were and made a payment. Also, you made the payment knowing what you were paying for. But I am not sure what you are saying. Did you pay money to the school organization for materials you needed as a member of the organization? Or did you act on behalf of the school organization and buy materials from an outside store or vendor? Usually if a person is buying items on a no refunds basis, then those are the terms if notice of those terms was given, unless there are some other circumstances, such as if the items were not delivered in usable condition.

You made an offer ($175.00) to a company knowing that refunds were not permitted. I assume the company cashed your check or ran your credit card and sent you the things you ordered. At this point the contract is completed and both parties must live with the terms they agreed to. A signed contract is not necessary to bind each party to the deal. If you are a minor, you might be able to get the company to refund your money on the grounds that minors usually aren't considered competent to agree to contracts.