The RA are quite notorious for taking a long time to reply, so I doubt this is a technical issue. Whether you wait or not is up to you. I don't know what the RA is like once you're in, as I've never worked with them.
You might want to try IRC. I believe they have a channel called #republica but you can check the /list to make sure.
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."
If you applied you will have to be cleared by CRI. That may take a few days.
From the faction description:
“To sign up with the Alliance to Restore the Republic, apply via Darkness, and be prepared to give your history and information to CRI. Once you are cleared by CRI, you will get the Academy and joining Information you need, please be patient. New members, please refrain from choosing your characters starting position until assignment.”
I'll add that we've been infiltrated by legit spies in the past, & this pushed us to have stricter controls of who enters(any comparison with real life would be enlightening - some real life country has been infiltrated & attacked some years ago, then responded by increasing its entrance controls.....).
Welcome to the RA, by the way. Unfortunately, I'm not in position to help you any more.