At present this is no public database where you can query an MID (to the best of my knowledge). You may search Google to see if someone else might have had any interaction with the MID in question and posted it on internet, which Google indexed.

Barring this, the only way is to ask your bank. Dispute the transaction or say you don’t remember doing it and need to know who the merchant is. Your bank should easily be able to sort this out for you.