Actually its far better and easier if you can show up at the station early enough and have the agent reissue your tickets. If you don’t, while you can still get credit for the trip it will take you a lot longer to do so and require more effort on your part. A simple phone call will NOT get you credit.
Normally when the conductor turns in the part of the ticket that he keeps, as that is scanned into the computer it will record your points for the trip if your Guest Rewards (GR) number is printed on the ticket. It normally takes about a week or two for your points to show up on your account.
If your GR number is not on the tickets at that time, it will require far more work on your part to get credit for the mileage. It will not be done automatically.
If you can't get the tickets changed before the start of your trip, then once you've returned you will have to Xerox your ticket stub. Then you will need to mail that copy, along with a letter including your GR number and address, to Amtrak in order to get credit. This process is also much slower, typically taking at least 2 months before you will receive credit. A friend of mine actually had to submit one of his trips 3 times over the course of about 7 months before he finally got credit.
So if at all possible, I would highly recommend trying to get your tickets reissued before you start the trip.
Regardless, have a great trip!