SAN FRANCISCO—Supporters of Vermont Sen. Bernie Sanders are hoping to capture the California Democratic Party’s top leadership post this weekend, rallying around an insurgent candidate who has promised to reorder the party establishment and redefine what it means to be a Democrat in this deeply blue state.

Delegates to the state Democratic Party convention in Sacramento will choose a replacement for departing chair John Burton, who has been credited with helping solidify the party’s control of the state, even at a time when...