Vallejo to San Francisco without toll?

Yes it's possible, but it's lengthy. You basically have to head south to Milpitas, take 237 E to 101 N and then drive up the peninsula on 101 to San Francisco. Depending on your southbound route from Vallejo, you don't have to pay toll to cross either the Carquinas bridge or the Benicia-Martinez bridge in the direction of travel.

You can easily drive back to Vallejo from San Francisco without paying tolls by taking the Golden Gate Bridge.

If you take I-80 down to I-580, into the East Bay, you won't pay a toll going over the Carcenas Bridge but, you will on the way back. Otherwise to avoid toll on the way back from San Francisco, you can Go across the Golden Gate Bridge and take HWY 101 North to HWY 37 which will take you into Vallejo.

Yes if you go across the GG Bridge north to Hwy 37, take that to Vallejo. No tolls.