While at CBS, Herrmann met Orson Welles, and wrote or arranged scores for radio shows in which Welles appeared or wrote, such as the Columbia Workshop, Welles's Mercury Theatre on the Air and Campbell Playhouse series (1938-1940), which were radio adaptations of literature and film. He conducted the live performances, including Welles's famous adaptation of H. G. Wells's The War of the Worlds broadcast on October 30, 1938, which consisted entirely of pre-existing music. Herrmann used large sections of his score for the inaugural broadcast of The Campbell Playhouse, an adaptation of Rebecca, for the feature film Jane Eyre (1943), the third film in which Welles starred.  When Welles gained his RKO Pictures contract, Herrmann worked for him. He wrote his first film score for Citizen Kane (1941) and received an Academy Award nomination for Best Score of a Dramatic Picture. He composed the score for Welles's second film, The Magnificent Ambersons (1942); like the film itself, the music was heavily edited by the studio, RKO Pictures. When more than half of his score was removed from the soundtrack, Herrmann bitterly severed his ties with the film and promised legal action if his name were not removed from the credits.  Herrmann also created the music for Welles's CBS radio series the Orson Welles Show (1941-1942), which included the debut of his wife Lucille Fletcher's suspense classic, The Hitch-Hiker; Ceiling Unlimited (1942), a program conceived to glorify the aviation industry and dramatize its role in World War II; and The Mercury Summer Theatre on the Air (1946). "Benny Herrmann was an intimate member of the family," Welles told filmmaker Peter Bogdanovich.  Herrmann was among those who rebutted the charges Pauline Kael made in her 1971 essay "Raising Kane", in which she revived controversy over the authorship of the screenplay for Citizen Kane and denigrated Welles's contributions.

Answer this question "When did they start working together?" by extracting the answer from the text above.
When Welles gained his RKO Pictures contract, Herrmann worked for him. He wrote his first film score for Citizen Kane (1941)