I have a business customer who does business with a money transmission/wire service, Intermex. https://corporate.intermexonline.com/en/
I believe the customer is working as an MSB agent for Intermex. The customer is depositing the cash through his business account through our bank but it appears that he's working as the agent for Intermex. I understand that an MSB agent does not have to register as an MSB but is that even if they're running their cash through our bank? Am I wrong and the customer is an actual MSB?

It comes down to whether they are an agent or not. Ask them for a copy of their agency agreement, if they are indicating they are an agent. Who they run the cash through is not relevant.