Buy lengths of studding of the appropriate thread – which would seem 2BA rather than 3/16″BSF from that parts-list – and make some screws. On machines that old they were typically just rounded off at each end, and given a fine screwdriver slot.
I don’t know if the length is to allow a lock-nut.
Don’t try to modify the thread. That is likely only to end badly; and anyway the more consistent with the rest of the machine the better.
Besides, there is no 3/16″ UNC or UNF size, but the nearest, 10-32UNF, has a crest diameter of 0.190″. So whilst a nominal replacement for 3/16″ BSF the fit of the screw is likely to be poor. A gib grub-screw does not take much load but you don’t want them possibly slacking off in service, and anyway the work would be a bodge!
Verify the thread and buy the right material: try someone like EKP Supplies.
(Incidentally I have a very similar task to complete, for a boring-table made to fit a Harrison lathe, and having long 2BA gib screws.)
If you do need cut the threads, suitable dies are available too: try, e.g. Tracy Tools.
Note though that the crest diameter of 2BA is 0.185 inch so 3/16″ (0.187″ ) dia rod to size may be a bit too tight to cut, and turning down that length won’t be easy with ordinary tooling. So using studding of correct thread to match what is there, is the best option.