Of course they can . . . because it's a "private association," which end-runs the whole idea of constitutional rights, which are designed to limit the power of government. Catholic associations don't have to admit non-Catholics, sororities don't have to admit men, etc. HOAs bastardize this by being "private" organizations that are fulfilling quasi-governmental-at-best roles.
HOAs are a creature of state law and only have power to the extent state law confers it. They are not voluntary associations of the sort contemplated by the First Amendment. You can't choose to join an HOA unless you own land that belongs to it, and you can't choose not to be a member when you acquire a property.
Yes, many HOAs have rules against political signage. This is blatantly a first amendment violation, but since they're technically a private organization, they can do that.
It depends on the state. Many states restrict HOAs from promulgating or enforcing rules about political speech. For those that don't, and where an HOA goes overboard in enforcement, I would very much like to see the case tried in Federal court. A proper ruling on the Constitutionality of certain HOA covenants is long overdue.
