"What would make a major publisher buy a manuscript (or two, or three) from a new author and then take a year (or two, or three) to choose an illustrator? I'm wondering if the issue is needing a well-known illustrator for an unknown author, or something else. And on a related note, how often do publishers buy manuscripts but never publish them?"
Well, in my experience, one year is different from two.
Sometimes things just get in the way, and a project is shuffled from season to season before everyone settles down and makes the illustrator decison and the book's schedule gets figured out.
(And if an illustrator has been chosen, then there's often waiting if that illustrator is in demand. But that waiting happens after the publisher contracts with the illustrator.)
But two years before the illustrator is chosen is about as long as you want to wait before you start asking some diplomatic questions. Because yes, sometimes publishers buy manuscripts and then their read on the market changes in an important way, or the editor who was excited about the project leaves for another house... and the project gets cancelled. This is not something that happens a lot, and publishers don't like to do it, but printing each book represents a chunk of money roughly equal to our own yearly salaries, so we can get a mite nervous if we're no longer sure a book is going to work out.
If it has been two years for you, and no illustrator has been chosen, get in touch with your editor and ask her pleasantly (because there really may not be any cause for alarm) if there are any concerns about the project. If your editor feels the project is no longer serving the market as originally envisioned (perhaps another, very similar book has been published by another house, eg) then you can suggest that your contract be amended to apply to a new manuscript, to be mutually agreed upon. This is the sort of play that I see from agents, but almost never from authors, because agents know how things work better than most authors.
If your editor indicates that there is no hesitation about the project's marketability, suggest nicely that you're a bit worried that it may be losing some momentum at the house. Do not reference the failure-to-publish clause in your contract at this point, but be aware of it.
Do take it easy. (Though I know--it's hard.) Editors' timelines are always much longer than authors would like them to be. It's amazing how quickly time slips by when you're this busy all the time. And even if the project is not to be at that house, you still have options.
Acting pleasant and professional about whatever's going on at your publisher will endear you a great deal to your editor--and will reinforce the idea that they can work with you, and that they want to.