If you go to Pokemon.com, and log in to your Pokemon Trainer Club account or whatever it's called, you can print a new ID. It will have the same number that you previously had, as well as your name.
Once you log in, go to the tab that says "Play! Pokemon" and there's a big orange button near the top that says "Print a Player ID Card" and then you do a bunch of customization to your ID, and then you print it.
Hope I could help! I actually lost my ID a while ago, but my main Tournaments Organizers know my ID, so I didn't really need it, until I went to tournaments in different area's.