What are the names of the 10 largest golf handicap computation
services in the US; and how many members or particpants (golfers) do
each of them have?

This does not answer your question completely, however it may be some advice.
the largest service is Golf Handicap and Information Network, (GHIN).
There are currently 59 associations, over 10,000 clubs and 1.9 million
golfers throughout the United States that make up the GHIN service.

Thank you very much. There are about 8 or 10 other computation
services, one I think is called handi-com (not sure) -- i am looking
for the names and potentially point of contact with those services