Sorry if it is confusing.
What does a Starbuck serve?
What do Starbucks serve?
The second one makes sense for me, since there are many Starbuck shops, but I'm still uncertain.
Which one should be grammatically correct?
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The name of the company is Starbucks -- it is never written without the 's', whether you are speaking of the singular company, a single shop, or multiple shops.
If you are speaking of a single shop, use a singular verb:
What does the Starbucks on the corner serve?
If you are speaking of multiple shops, use a plural verb:
What do the Starbucks in Minneapolis serve that other Starbucks don't serve?
If you are speaking of the company, AmE use calls for a singular verb:
What does Starbucks permit its stores to serve?
I would actually just say
What does Starbucks serve?
I think this, because Starbucks is a singular noun. Starbucks is a chain of stores. I think typically it is unnecessary to ask about one particular store, because a chain of stores all serve the same thing (almost).
I would think if someone asked about just one store that they were asking what that store serves that is different from all the other stores.