Ping frictions are the only ones I've found with a very thin shaft, which will fit the small holes on many vintage ukes. However, the Ping bushings usually require opening up the top headstock holes. But since many old ukes just used washers, not bushings on top, you can use washers instead, often the same ones that came on the vintage uke.
Replacement screws are almost certainly available somewhere but you might struggle to identify the existing screw. As a start point you need to accurately measure the screw’s diameter and count how many teeth it has in one inch of its length. With that information it’s possible to suggest what to ask specialist stockists for.