"Commonly accepted" does not mean the same thing as "legal." If the HOA agreement allowed for the use of proxies, then having done so would be fine--decisions made with proxies votes would be valid. Similarly, if the quorum language only required half plus 1 of owners in residence, then its likely you have valid quorums. But HOA agreements are enforced according to their language: if a quorum of all homeowners (half plus 1) was required to transact business and there was no provision for proxies, then yes, this family would have a good case that HOA business was not conducted legally.