HOAs are frustrating. I would suggest reading the ordinances and covenants. Everyone receives a copy when they purchase property. If she doesn't have a copy, ask the HOA how to obtain one. There may be a way to get them to change it if they have a majority of the members to vote to do so, but it depends on what the ordinance says. Good Luck. It won't be easy.
HOAs, can represent, the Despotizm of intrencht Majority Apathy and Ignorancy. I would advise, to find out who the People are, and become knowen to them, as her Neighbor. Later on, advise them of y'r chiCken Desire. If they know you are Responsible and trust Worthy, it will be easier. I recommend Patiency and about one Year.
The covenants of an HOA are a contract between the association and each member. It is VERY difficult to change the CC&Rs; deliberately so. The CC&Rs are the equivalent of the constitution, not simply a set of easily voted out rules. Making changes to the CC&Rs usually requires approval of either 66% (2/3rd) or 75% (3/4th)of all members. Not a percentage of the votes received, a percentage of the total number of members. And unless something is very controversial, you will not usually get anywhere near that large of a percentage of total votes--getting whatever the required quorum is is nearly always difficult.
If the restriction against chickens is a board approved rule rather than a CC&R covenant, you merely have to convince a majority of the board members.