This LP does not provide any credits for the soundtrack composer, and in some contemporary sources the credit goes to Morris Stoloff, who is listed as directing the orchestra on the jacket. But the film shows up in George Duning's filmreference.com resume, so let's assume he actually was the one who did the score.
As a practical matter, it doesn't make a whole lot of difference, because you don't hear much of the actual score on this record. What you do hear are a few songs by Lester Lee with lyrics by Ned Washington. The tune about the Marines has words by Allan Roberts. Jo Ann Greer dubbed the vocals for Hayworth. The harmonica player is Leo Diamond