To my knowledge, Doris Day was never a theater performer.
However, in 1944, "Doris Day" was listed in the cast of "Tangled Web" a pre-Broadway play that closed in Wilmington.
Doris would have been 22 at the time. She changed her name in1939 and started recording in 1945, so it could have been her. Incidentally, Morton daCosta was also in the play.
I don't believe she ever spoke of a theatrical career and I don't know if this was listed in her resume. Was this in fact the Doris Day we know that appeared in "Tangled Web" or any other theatrical performance?