How do I figure out who issued a credit card from just the first 4 digits?

Here's a wikipedia page which unfortunately was deleted and is out of date. You can look for other sites that will allow credit card merchant lookups, but most that I'm aware of required paid access. Good luck getting your refund.

Ask the issuer

If you don't know the numbers, then the only one who'll know it will be the institution/bank who issued that card. It's likely that the merchant itself has no more information than they gave you, as it's recommended to not store the full card number and they can do a refund based on the transaction reference IDs.

The first six digits of the card are known as the BIN (or IIN). They can be used to identify who issued the card, and sometimes what type of card it is. This lookup service would help, but it requires all six. Just the first four won't work unless you want to try all 100 possibilities. Even with that, you only might be able to tell what bank issued the card, which is something you should probably be able to remember in the first place.

If the merchant issued a refund to what is effectively the wrong card, because they didn't bother asking you for your current card (the one they had on file was likely expired anyway, if it's that old), then it's up to them to get that money back and reissue your refund. If, however, it's an account you still have open in your name (even if you don't have the physical card) the merchant is in the clear and you have to do your homework with old statements or credit reports to track down that account.

Check your email or your snail mail for a statement or message telling you the last four digits of your old cards. These messages would have been telling you a new statement was available, or that a new card was being issued.

You were either transferring money to pay the bill or paying by check. That means the check or electronic transfer would have left a trace in your bank account. Check your bank website or account statements for payments to credit cards. There might be enough information to be able to identify the correct card.

If you didn't get a paper monthly statement you would have been logging into a website, and the login still might work.