This is yet another crappy and ridiculous claim made in Daniel Cassidy’s book of fake Irish derivations for English terms, How The Irish Invented Slang. I don’t know much about this word, heeler, but apparently in American politics a ward heeler was the assistant to a politician and made sure that the grass roots support in the ward was kept sweet. This is obviously a reference to his walking around at the heels of the politician or perhaps his role bringing the electorate to heel.
According to Cassidy, it comes from the Irish word éilitheoir. This word is defined by Ó Dónaill as claimant or as plaintiff, neither of which is very appropriate. It is the clear mark of a crank that they walk past an obvious and believable explanation in favour of a contrived and unbelievable one, and Cassidy did this all the time.