On the left side under Advanced Search is a box for Element ID. I think that should work. Assuming of course all the parts you need are available though the Shop@Home pick-a-brick system.
Option 2: scan or photograph the inventory portion of the ATAT instructions and note which items you're missing. I'm sure we could help you find those on Bricklink.
Alternate approach: you could try contacting Lego customer service by email or through the website and see if they could hook you up with the necessary replacement parts. I'm iffy on that approach because I don't think they should be responsible for incomplete second hand sets and I bet they'd agree. But who knows. At the very least, and assuming you already have a list of the missing part numbers, you could ask if those were available to purchase. Maybe they have a quicker way to check the PAB database than doing it yourself manually. That's just my lazy side being optimistic though. Props to them if they do the legwork for you but I'd wager you'll get a response that's closer to my first suggestion.