I can leave Rome at 8 and arrive in Pisa at 11. I'd like to take the train to La Spezia/Vernazza at 3. Would that leave me enough time to take in the tower? Can I store my bags at the train station? I don't want to be too ambitious!
Yes, definitely do-able. It's about a 30-minute walk from the train station to the Field of Miracles. I'm not sure if you will have time to climb the tower though (maybe if you perfectly time your reservation). And I think Pisa Centrale is one of the last stations that has a place to store bags. Check your bag at the Deposito Bagagli office (near Track 1, if memory serves) with a person and don't use the lockers. Then just be sure to give yourself enough time to walk back to the station and enough time to retrieve your bag (the lines can get long). I can't remember if the Deposito Bagagli office closes for lunch (potentially for 2 hours) or if it's always manned, so you may want to double-check when you drop off your bag.
You will have enough time! Honestly, there's not a ton to do in Pisa so four hours should be plenty. And there is a left luggage station, so you should be fine on that front as well!