How many LGBT Republicans are there in the US?

The entire Republican party has hundreds of openly gay politicians. There is no master list.

If you're talking about Congress, there have been 7 openly gay members as of 2017:

1. Stewart McKinney (1931–1987) Connecticut Republican 1971–1987
   
2. Bauman Robert Bauman (born 1937) Maryland Republican 1973–1981 Came out after his time in Congress
3. Hinson Jon Hinson (1942–1995) Mississippi Republican 1979–1981 Came out after his time in Congress
4. Gunderson Steve Gunderson (born 1951) Wisconsin Republican 1981–1997 Outed on the floor of the House in 1994, and became the first openly gay Republican representative.
5. Kolbe Jim Kolbe (born 1942) Arizona Republican 1985–2007 Came out in 1996 after voting for the Defense of Marriage Act, and was the first openly gay person to address the Republican National Convention. He was the second openly gay Republican to serve in Congress.
6. Huffington Michael Huffington (born 1947) California Republican 1993–1995 Came out as bisexual in 1998, the first bisexual to have been elected to Congress.
7. Foley Mark Foley (born 1954) Florida Republican 1995–2006 Came out after congressional page incidents.