We deal with these issues at work from time to time. Mostly medical/financial stuff.
What makes me question this sort of robot is two things: our classy printers support secure storage and local release by password or token(you can also buy software to do this at the print server level but we've avoided that) and we can put a good-enough, reliable-enough laser on your desk for ~$200(maybe half that if you skimp on reliability, more like $400-450 for something pleasingly overqualified).
This forces the robot option to be quite cheap and reliable(especially since it can only be shared as fast as it can move between users, so multiple bots will be needed in busier situations.) is that impossible? Probably not. Is that going to take some serious refinement to even get close to just throwing brute force and cheap hardware at the problem? Oh yes, yes it is.