Police were alerted about the robbery at 2:26 a.m. The suspects had assault weapons and held up the security guard, who was not injured. The men took off in another vehicle and left the SUV inside the bank.
Police are looking for three men in a white Ford Taurus who were last seen heading eastbound across the Bay Bridge.
The museum's historic stagecoaches went undamaged during the robbery.
Wells Fargo officials said the museum will reopen.