Answer by taking a quote from the following article:

Annie Get Your Gun is a musical with lyrics and music by Irving Berlin and a book by Dorothy Fields and her brother Herbert Fields. The story is a fictionalized version of the life of Annie Oakley (1860-1926), a sharpshooter who starred in Buffalo Bill's Wild West, and her romance with sharpshooter Frank E. Butler (1847-1926). The 1946 Broadway production was a hit, and the musical had long runs in both New York (1,147 performances) and London, spawning revivals, a 1950 film version and television versions. Songs that became hits include "There's No Business Like Show Business", "Doin' What Comes Natur'lly", "You Can't Get a Man with a Gun",

When the traveling Buffalo Bill's Wild West show visits Cincinnati, Ohio ("Colonel Buffalo Bill"SS), Frank Butler, the show's handsome, womanizing star ("I'm a Bad, Bad, Man"SS), challenges anyone in town to a shooting match. Foster Wilson, a local hotel owner, doesn't appreciate the Wild West show taking over his hotel, so Frank gives him a side bet of one hundred dollars on the match. Annie Oakley enters and shoots a bird off Dolly Tate's hat, and then explains her simple backwoods ways to Wilson with the help of her siblings ("Doin' What Comes Natur'lly"). When Wilson learns she's a brilliant shot, he enters her in the shooting match against Frank Butler.  While waiting for the match to start, Annie meets Frank Butler and is instantly smitten with him, not knowing he will be her opponent. When she asks Frank if he likes her, Frank explains that the girl he wants will "wear satin... and smell of cologne" ("The Girl That I Marry"). The rough and naive Annie comically laments that "You Can't Get a Man with a Gun." At the shooting match, Annie finds out that Frank is the "big swollen-headed stiff" from the Wild West show. She wins the contest, and Buffalo Bill and Charlie Davenport, the show's manager, invite Annie to join the Wild West Show. Annie agrees because she loves Frank even though she has no idea what "show business" is. Frank, Charlie, Buffalo Bill, and everyone explain that "There's No Business Like Show Business."  Over the course of working together, Frank becomes enamored of the plain-spoken, honest, tomboyish Annie and, as they travel to Minneapolis, Minnesota on a train, he explains to her what "love" is ("They Say It's Wonderful"). Buffalo Bill and Charlie discover that their rival, Pawnee Bill's Far East Show, will be playing in Saint Paul, Minnesota while the Wild West show plays in nearby Minneapolis. They ask Annie to do a special shooting stunt on a motorcycle to draw Pawnee Bill's business away. Annie agrees because the trick will surprise Frank. She sings her siblings to sleep with the "Moonshine Lullaby."  As Annie and Frank prepare for the show, Frank plans to propose to Annie after the show and then ruefully admits that "My Defenses Are Down". When Annie performs her trick and becomes a star, Chief Sitting Bull adopts her into the Sioux tribe ("I'm An Indian Too"SS). Hurt and angry, Frank walks out on Annie and the show, joining the competing Pawnee Bill's show.

Did Frank and Annie Oakley have a shooting match?
Annie meets Frank Butler and is instantly smitten with him, not knowing he will be her opponent.