Shrug, a "neutron howitzer" isn't uncommon in physics labs -- you're going to need a fair amount of paperwork to approve it though. Mind, those usually use alpha+beryllium reaction, not fission.
Want to say, research reactors these days are always more featureful than that, but new reactors aren't exactly an everyday occurrence these days, let alone publicized, so, no idea. Could very well be there's some lab somewhere, that's recently moved into nuclear physics and built one like that as a graduate project; wouldn't really need to order one whole (if indeed you could still find such), just procure the specialty materials.
Cool fact, my alma mater had (likely still has) a lump of plutonium in a handsome (classic 40s-50s hammertone) enclosure, with a Los Alamos placard neatly riveted to it. IIRC it came to something like 150mC or 50g. You'd need more than a few such units smushed together to get anywhere near criticality.
Tim