While you are on a customer service call, how would you clarify the name? Is it grammatically correct to say, " let me confirm your name"?
"Let me confirm your name" is grammatically correct; it is also an appropriate question when the person has already said their name but you want to be sure that you've heard it correctly, or if you need to transcribe it accurately.
You could also try "Please remind me of your name" if, at the same time that you want find out what the other person is called, you wish to politely imply that you are to blame for the fact that they are so forgettable. :)
As previously commented, simply "Let me confirm your name" doesn't sound like a question and the caller probably would catch on the to the fact that you would like a response (e.g. that you are trying to clarify their name.)
A more suitable example depends on how formal you are and the specific circumstances, but could be:
Could you remind me [the spelling of] your name please?
"Could I just confirm your name: [Joe Bloggs? J-o-e, etc.]"