There is a technique for turning small ball shapes which might work for this. You take a piece of silver steel rod a little larger than the ball size, and drill a hole in the end a little smaller than the ball size.Face the end and then harden. Fit the other end into a file handle, and arrange a rest in the tool post close to the job. Having roughly formed the ball end with a 45 degree chamfer, use the hand turning tool to form the ball, moving it around the end of the job so that it can cut. The circular cutting edge on the inside of the hole will form the ball. The outside edge is not a cutting edge and should never contact the job.
I've used this technique to form ball ends on tiny handles as small as about 1/8 inch or so. A refinement is if you can manage to face the end back at an angle to give some relief to the cutting edge. This sort of tool will work for a range of sizes, the smallest being the size of the hole.
Otherwise, you can make a form tool from gauge plate by drilling a hole the size you want and then cutting away the bits you don't want to leave a half or quarter circle cutting edge.
Case hardening should be enough for the tappets for a vintage style of engine.
regards
John