Does my membership in the HOA affect my ability to moonlight?

The easy answer -- Yes and No.  

The more detailed answer -- If you are a resident or a fellow, with some very unique exceptions for a few of the non-ACGME accredited programs, you are classified by the medical staff bylaws as a CPT (clinical program trainee).  All CPTs work for UM based on the contract negotiated by the HOA, regardless of official "membership" in the HOA.  So, even if you elect to rescind your HOA membership you are still under the same contract with the University (just without all of the benefits and protections of being part of the HOA).  The eligibility and process rules surrounding moonlighting remain the same for all CPTs regardless of HOA membership -- this includes 80 hour work week restrictions, completion of eligibility forms, etc.

Separate from the HOA, the ACGME has certain restrictions on moonlighting with regards to what services a resident/fellow can provide as a moonlighter if they are still in training.  The language from the ACGME, having nothing to do with the HOA or even UM specifically, restricts moonlighting in the same specific specialty of your training program.  

That's not to say that anesthesiologists can't moonlight in any area of anesthesiology; rather, pain fellows can't moonlight to cover the pain service [same specific specialty training], but critical care fellows can moonlight to cover GA cases in the ORs [different specialty components of anesthesiology].  All moonlighting is still subject to program director approval.