I know that I and a lot of others were, for some reason, did not receive instant approval for the new Chase AGR card. I think it took about two weeks for me to get the card from when I applied.

That being said, I would not wait to purchase tickets if I already knew when I was going to travel. Amtrak uses fare buckets and if you wait, the fare you will need to pay will increase.

What you could do is buy a ticket now online--don't pick it up at a ticket machine yet, though. If you get the AGR credit card and the fares are still the same, you can buy a new ticket using the AGR credit card and then cancel the first ticket online (free).