What can I do if a company won't refund me without proof of purchase?

Don't just accept "no proof, no refund" as final - companies often have other ways to verify purchases. Check your email for order confirmations or receipts they may have sent, dig through your bank or credit card statements for the transaction, or ask if they can look it up by your account or phone number. If they still refuse and you paid by card, you can file a chargeback with your bank/credit card company as a last resort, though that should be your nuclear option since it can burn bridges with the company.

Most companies can pull up your purchase using your credit card number, email address, or phone number on file - you don't actually need the physical receipt! I bought yarn from a shop online years ago and lost the confirmation email, but when I called they found the order in seconds just by having me verify my card details and the approximate date. If they're still refusing, ask specifically what information they need to locate the transaction in their system, and escalate to a supervisor if the first person won't budge. You can also file a chargeback with your credit card company if nothing else works - that tends to get companies' attention really fast!