Unfortunately for the original poster I'm on the side of the HOA, at least for now. Contract law is older than government, not just the government of Iowa, or the USA, but the entire concept of a government. A government is nothing but an extension of the concept of a contract, it is a contract among the people. If the HOA has a policy on firearms, and you signed it, then you have bound yourself to its rules. If that means showing a permit then that means you show your permit.
In Cedar Rapids there is a monthly open carry night at Godfather's Pizza in the Town and Country Shopping Center, to enter with your firearm you must show your permit. Private property owners, under state law, may permit people to enter armed, or not, or place conditions on your being armed. That may include having to show a state issued permit. With a HOA the HOA board retains partial ownership of your property. Partial ownership of property is another concept that has long standing in law. As they retain partial ownership they have a say on what you can do on "your" land, that includes having to show a permit to carry your Kimber openly as you walk to and from your mailbox.
I have not read the HOA agreement you signed, obviously, so I don't know what you agreed to when you rented your property. Unless you can show that the HOA do not have a say on the open carry of weapons then I would assume that they do retain the right to demand to see your permit. You might be able to argue otherwise in court but I don't see a whole lot of precedence in your favor.
Of course that is my opinion, and it's worth exactly what you paid for it.
