Merchant of record
The merchant of record is the party legally selling to the shopper: the name on the receipt, the entity that owes the sales tax, takes the chargeback and carries the refund obligation.
In a normal online sale this is the store, and the question never comes up. Agent mediated purchases raise it, because a third party may sit between the shopper and the merchant, collecting the payment and passing on the order.
The answer is not the same everywhere and it is being settled commercially rather than technically. What is worth knowing is that it is a real question with real tax and dispute consequences, not a piece of jargon.
Why it matters to a store
If you sell through an agent facing surface, find out who the merchant of record is before the first chargeback tells you.