It is our Credit Union policy to only bank US Citizens and Resident Aliens. Our CIP policy states that an ITIN number can be used for Resident Aliens who are not eligible for an SSN. I just want to confirm that Resident Aliens may be issued an ITIN and may not have an SSN, correct? Also, if the Resident Alien does not have a green card, but states they meet the physical presence test, how is your financial institution going about verifying that?

You have created a situation in which you are falling on your own sword. It is your policy, and you need to decide how you want to police it. There is no way to verify under the substantial presence test. Chances are that an alien that meets the substantial presence test is either here on a visa or is here illegally. So, you have to ask yourself, what is the actual goal of the organization by having this policy that cannot be specifically implemented and verified.