In your CIP process, the LLC will be your customer (under the regulation). You obtain the name of the LLC, its street address, its TIN (which is its member's SSN). You need to verify the identity (existence) of the LLC.
Your bank's CIP may (probably should) also require you to obtain CIP information on the individual who is the single member of the LLC. That will include his/her name, SSN, DOB and street address. Of course, you'll verify his/her identity.
If the LLC ever changes to a different status -- if it's no longer a single-member LLC, for example -- it has to get its own EIN and you'll use it.
John S. Burnett
Fighting for Compliance since 1976
Bankers' Threads User #8