According to Robert John McCrary, chess historian:

"In 1938 Caroline Marshall, wife of long-time US Champion Frank Marshall, organized a US Women's Chess Championship at the Rockefeller Center in New York City."

My finding is a bit different, but I'm not sure what the difference means.

To the best of my knowledge, and contrary to most of the places I've looked to piece this together, this information is correct and complete. I don't know the reasons to the blank dates, but it seems there were no championships held those years and I presume that the last champion was considered the current champion for those years. If anyone finds an error or anything missing, I'd very much appreciate hearing for you. 

1937        Adele Rivero
1938        Mona May Karff
1939  
1940        Adele Rivero
1941        Mona May Karff
1942        Mona May Karff
1943
1944        Gisela Kahn Gresser
1945
1946        Mona May Karff
1947
1948        Gisela Kahn Gresser - Mona May Karff
1949
1950
1951        Mary Bain
1952    
1953        Mona May Karff
1954        
1955        Gisela Kahn Gresser - Nancy Roos
1956
1957        Gisela Kahn Gresser - Sonja Graf
1958
1959        Lisa Lane
1960
1961
1962        Gisela Kahn Gresser
1963
1964        Sonja Graf
1965        Gisela Kahn Gresser
1966        Gisela Kahn Gresser - Lisa Lane
1967        Gisela Kahn Gresser
1968
1969        Gisela Kahn Gresser
1970
1971
1972        Eva Aronson - Marilyn Koput
1973
1974        Mona May Karff
1975        Diane Savereide
1976        Diane Savereide
1977        Diane Savereide - Rachel Crotto
1978        Diane Savereide - Rachel Crotto
1979        Rachel Crotto
1980
1981        Diane Savereide
1982
1983
1984        Diane Savereide
1985
1986        Inna Izrailov
1987        Anna Achsharumova
1988
1989        Dr. Alexey (Rudolph) Root
1990        Elena Donaldson
1991        Esther Epstein - Irina Levitina
1992        Irina Levitina
1993        Elena Donaldson - Irina Levitina
1994        Elena Donaldson
1995        Anjelina Belakovskaia - Sharon Burtman
1996        Anjelina Belakovskaia
1997        Esther Epstein         
1998        Irena Krush
1999        Anjelina Belakovskaia
2000        Elina Groberman - Camilla Baginskaite
2001/02   Jennifer Shahade
2003        Anna (Kahn) Hahn
2004        Jennifer Shahade
2005        Rusa Goletiani

Here's a game played by two of the pioneer women champions in 1938

[Event "US Women's Championship"]
[Site "?"]
[Date "1938.??.??"]
[Round "-"]
[White "Adele Rivero"]
[Black "Mona Karff"]
[Result "0-1"]

1. d4 f5 2. g3 e6 3. Bg2 Nf6 4. c4 c6 5. Nc3 d5 6. Nf3 Bd6 7. Ne5 O-O 8.
O-O Nbd7 9. f4 Ne4 10. Nxe4 fxe4 11. Qb3 Qe8 12. cxd5 cxd5 13. Bd2 Bxe5 14. fxe5 a5 15. Rxf8+ Qxf8 16. Rf1 Qe7 17. Qb5 b6 18. Qc6 Rb8 19. Qc7 Ba6 20. Rf2 Rc8 21. Qa7 Bb5 22. Qb7 Qe8 23. Bh3 Rc6 24. b3 Qb8 25. Qxb8+ Nxb8 26. Bg5 h6 27. Be7 Kh7 28. Bd6 Rc1+ 29. Kg2 e3 30. Rf1 Rc2 31. Bxe6 Rxe2+ 32. Kf3 Nc6 33. Bf5+ g6 34. Bc8 Nxd4+ 35. Kg4 Rf2 36. Rc1 h5+ 37. Kh3 Bf1+ 38. Kh4 Nf3#
0-1