My wife has a consulting business that she owns. If she makes me an agent of her business will Restaurant Depot let me get a membership even though it's not food related?
Unfortunately, this is heavily dependent on the person that process your application. When I joined you could get a temporary membership online, then you would go in a store to get it finalized.
Typically, in the store, they do not ask questions about the nature of the business, but are pretty strict in requiring documentation regarding the actual operation of the business, especially a federal Tax ID or other evidence.
With that in mind, a client is a client, so I would try to go with your wife's business. As the prior poster said, with a Tax ID they will likely let you in. No harm trying...
BTW their S. Marzano tomatoes are excellent and the price is pretty good. It comes in a big can, so I jar it in smaller containers and refrigerate.