If you know anyone in the medical field, or scientific field, you can
use latex tubing used for torniquets or the tubing from IV sets. Cut
it longer tan the current belt, and just tie it with a tight square
knot. It should transmit power for a small drill press. A stronger
solution may be to use the small tubing used for air in aquariums if
you sew it together (sharpen the end of a needle with a file, use a
strong string or fishing line to thread with, a pair of pliers to
push the needle through, and sew all the way around the outside of
the diameter). Here is a link to a picture of how you should sew it,
it is medically referred to as an end to end anastomosis used to sew
veins (or other tubular organ) together.
It will be difficult to pierce the tubing, it may be easier to first
use a heated sewing needle to make multiple holes through which you
can thread your string.
O ring belts can also be made easily, but finding large enough O
Ring may be difficult. I have glued it together with super glue
before. The tube belt I describe above will last longer though, and
likely be less expensive.
3/16" is not a standard V belt, as the smallest made commercially is
1/4" wide, and the smallest length I can find in that is 10"
(circumfrence around the outside of the belt) If there is a lot of
adjustment capability for the length, it may work. The trade number
you are searching for is a 2L100 belt, if you google 2L100 belt you
may be able to find a local supplier. It may be possible that you can
have it ordered at an auto parts store. Again, the tube belt may be
your cheapest and most reliable option. Plus you never have to wait
for repair parts if it breaks again!