[derobert]

That's basically what you do—you send the correspondence via certified mail, and keep the certified mail receipt. That doesn't prove what you mailed, but it gives verification that you sent something. (You purchased something from Equifax on the 1st, sent them a certified letter on the 10th. You claim it was the arbitration opt-out; they had better have some evidence it wasn't, or the judge/jury is probably going to believe you.)

Honestly for this (the Equifax thing), you just keep record of when you sent it—it's only an issue if you litigate, and then I'd expect your record of when you sent it + your testimony would be sufficient. But IANAL, and you should of course talk to one if it matters.

For your HOA, hopefully you have some record of when you sent the request (e.g., you kept a copy of the ARC application with a note that you mailed it on $DATE). (Of course, the HOA should be maintaining records of when applications are received.) Depending on what it is, this is something that may be worth paying for legal advice on.