Janet Gaynor -- long claimed to have been lovers with Mary Martin -- won Best Actress at the very first Academy Awards (1927/28).
For Best Actor, I suspect we'd have to go with Charles Laughton, who won Best Actor in the 1932/33 race for "The Private Life of Henry VIII," co-starring his long-time beard, Elsa Lanchester.
r8 is correct: Janet Gaynor.
She and her frequent screen partner, Charles Farrell, were both known to be gay in Hollywood circles, and many jokes were made about them at their expense.