Out of Left Field by Ellen Klages

Every boy in the neighborhood knows Katy Gordon is their best pitcher, even though she's a girl. But when she tries out for Little League, it's a whole different story. Girls are not eligible, period. It is a boy's game and always has been. It's not fair, and Katy's going to fight back. Inspired by what she's learning about civil rights in school, she sets out to prove that she's not the only girl who plays baseball. With the help of friendly librarians and some tenacious research skills, Katy discovers the forgotten history of female ball players. Why does no one know about them? Where are they now? And how can one ten-year-old change people’s minds about what girls can do?
 
Set in 1957—the world of Sputnik and Leave It to Beaver, saddle shoes and "Heartbreak Hotel"—Out of Left Field is both a detailed picture of a fascinating historic period and a timelessly inspiring story about standing up for equality at any age.