Welcome to the forum! Automatons are cool, would be neat to see what you come up with. There's an art museum near me that has a whole exhibit of wooden mechanical gadgets, that's my favorite part of their collection.
Mass produced consumer goods like this will use a lot of custom (or semi-custom) mechanical parts, so are are not likely to find any datasheets. You'll have to do some reverse engineering. The motor will almost certainly be either a simple brushed DC motor or a stepper motor. The former will have two leads, the latter will have 3-5 leads. There might be an encoder attached, which will have 3+ leads, or they might rely on the read head to provide position feedback to the motor relative to the data track. Beyond that, we'd probably need to see some pictures to help figure out what's going on.
FYI, I've seen similar mechanisms to what you're describing out of CD/DVD drives being sold separately on Amazon/ebay as small linear actuators. I can't find the right search terms now, but you might be able to find more of those mechanisms without having to cannibalize them yourself.