Car rental/ train from La Spezia

You don't really need a car in any of those locations. All are easily reached by train and those cities/towns have severe traffic restrictions in the historical center, therefore the car won't be of much use.
Cars are good only to visit the countryside of Tuscany or the small villages, if you plan to visit those, but certainly not Lucca or Siena, where the historical centers are largely off limits to cars (just like Vernazza).
However, since you will hopping on and off trains often, it's imperative that you learn how to travel light (carry on bag and one back pack only). Trains and heavy luggage don't mix well.
Lucca can be easily reached from Pisa on your return trip back from Vernazza.
Basically the train changes are:
Vernazza > La Spezia > Pisa > Lucca.
Then from Lucca to Siena you would take the train again:
Lucca > Pisa > Empoli > Siena
or also:
Lucca > Florence >Bus to Siena
The issue might be where to leave the luggage while visiting Lucca. Not sure if Lucca has a luggage storage depot at the station, but Pisa certainly does. So you could leave your stuff at Pisa station while you visit Lucca (just check the luggage storage hours at Pisa so that you retrieve the luggage before closing time).

If you absolutely must have a car after Vernazza, just go to Pisa by train. The Pisa airport has all the rental agencies you want and they are open on Sundays.

I would travel by train to Lucca, Pisa, and Siena, since car rental is limited on Sundays. If you need a car when departing Siena, then you could pick one up there. If Siena is not really your Sunday destination and you need the car to get to a rural agriturismo, then that would be a reason to get it in Pisa.

We just found out about the no rentals to as we are planning on doing the exact same thing but the following weekend. We are looking into renting a car on Sat. and leaving it overnight in the LaSpezia station garage and then driving to Tuscany on Sunday. Is anyone familiar with reserving a spot in the garage? Looking on line you can only do it a day ahead and they only have 116 spaces.
Also since it seems most of Italy is closed on Sunday can anyone recommend a nice place to stop for lunch between LaSpezia and Chianti?

Thanks for your thoughts,
Linda

We just found out about the no rentals too, as we are planning on doing the exact same thing but the following weekend. We are looking into renting a car on Sat. and leaving it overnight in the LaSpezia station garage and then driving to Tuscany on Sunday. Is anyone familiar with reserving a spot in the garage? Looking on line you can only do it a day ahead and they only have 116 spaces.
Also since it seems most of Italy is closed on Sunday can anyone recommend a nice place to stop for lunch between LaSpezia and Chianti?

Thanks for your thoughts,
Linda

Windowsdressing:

If Italy is closed down on Sundays, I recommend you pack some peanut butter and jelly before heading to the Chianti.

If you can't find any place that sells peanut butter, then stop at the Autogrills along the freeways. Autogrill is always open.

Thanks everyone for your suggestions. Linda I am thinking of doing what you are but renting on Friday and just parking til Sunday in a lot. If you find one please let me know.
We will have a fair bit of luggage as we are in Italy for over 2 weeks so I am concerned about hopping on and off trains and storage if we don't rent a car til later in the day on Sunday..

Julie

Italy is open on Sundays, but some services may be restricted or unavailable.

If you decided on renting a car, then these are your parking options.
1. La Spezia:
There is a parking structure at the train station.
http://www.mobpark.eu/en/park-centro-stazione-7.html
The parking garage is under the station, it's accessible from Via Fiume and has 218 car spaces.
The parking costs about €20 for 24h.
Below is a videoguide:
http://www.mobpark.eu/it/park-centro-stazione-6/info/videoguide.html
You can reserve a space here:
http://www.mobpark.it/ar/

2. Cinque Terre:
You could also drive to the Cinque Terre instead of La Spezia.
I stopped taking the train when the direct train Florence-Turin was re-routed via Bologna instead of via Genoa (after the high speed line was built). Those Express IC Florence-Turin trains allowed me to get to Monterosso from Florence with no changes, it was wonderful. Now you have to change two times at least (Pisa-La Spezia), so I drive.
However except for Monterosso-Fegina (the new part of Monterosso), the access roads to the towns are narrow.
I generally chose two towns to drive to. Either Riomaggiore (the closest to La Spezia), or Monterosso (the farthest).

To drive to Riomaggiore, once in La Spezia, you follow the directions to Portovenere, Cinque Terre. After you go around the port the directions split. Portovenere straight, Riomaggiore (Cinque Terre) up to the right. Once in Riomaggiore, at the entrance of town there is a parking structure (on the left).
The largest parking option is in Monterosso. To reach Monterosso don't exit at La Spezia, rather stay on the A12 freeway a little longer toward Genoa and exit at Levanto. Follow the directions to Levanto. Once in Levanto at the roundabout follow directions to Monterosso. At a certain point the road splits. One goes to the old Monterosso, one goes to the new Monterosso (Fegina). Take the latter. There is a large parking lot right on the beach.
Actually Levanto has parking too (including at the station). From there you would take the train back to the Cinque Terre (just minutes).
Don't attempt to go to any other town except for the above, unless you are a very experienced drive who can deal with negotiating a car with no more than a few inches on each side. The roads are very narrow.
Other details are below:
http://www.cinqueterre.eu.com/en/parking-cinque-terre