Where to reserve car rental - Pisa or Lucca?

I haven’t rented at either place but did a lot of research on it last year
Lucca would be fine for dropping off the car but the locations there are very small with short hours
Chances of getting a car large enough for your group , and an automatic if you need that- are pretty slim.
Airport locations will always have more choices.

I do see La Spezia mentioned here as a car rental pick up place, have you checked there?

We always use AutoEurope.com for our rentals
They are US based and are extremely helpful/ knowledgeable
You might give them a call toll free. Number is on their website
You can also check those Lucca locations on their website for hours/ car availability

Renting a car at Pisa airport is pretty straightforward but in your case it's probably more of a hassle than it's worth.
One alternative might be to take the train to Lucca (or Tassignano near Cappanori) and taxi to your accommodation from there. Since the rental car concessions in Lucca are closed from noon Saturday until Monday morning, you could just do without for those days and instead use the time to explore Lucca - a beautiful and atmospheric little town that would be a pleasant way to spend a Sunday afternoon.
Then on Monday morning have one or two of your party train into Lucca to collect the car, return to your accommodation to collect everyone else, and take off from there for your explorations of the surrounding area. One advantage of that approach is that you could probably opt for a slightly smaller (and cheaper) car since you'd only be transporting people and not their luggage as well.
Per Christine, do opt for a vehicle with automatic transmission - you can thank us later, and also go for the full insurance (via AutoEurope) for the peace of mind.
And don't forget your International Driving Permit.

And I agree with Robert especially since you cannot rent a car in Lucca and the Pisa airport is where you need to pick it up.

I agree. Call AutoEurope which is located in Maine.
The time we went to the CinqueTerre we arrived by rental car. Left it in parking lot there during our stay. It seems easier to get the car when leaving Venice, drop it off in Florence. Much easier than schlepping luggage on and off trains,

From the Cinque Terre the most logical place to pick up a rental car is in La Spezia. From there you can go to Capannori. Take the train from the Cinque Terre town you are staying to La Spezia. At the La Spezia Centrale station take a taxi to the rental office of your choice (most are near the cruise port).
If you are picking up on a Saturday, make sure you do so in the morning. Both Hertz and Europcar in La Spezia close Saturday afternoon at 12:30pm and Sunday all day. Avis-Budget is closed all day Saturday and Sunday, so it is not an option for you. Check with AutoEurope or Kemwel (same company now, as they merged). They are a reputable car rental broker I always use.
If you have a lot of luggage you might consider driving back to the town you are staying and pick the rest of the party. However the road to the cinque Terre towns is narrow so it depends on how comfortable you are behind the wheel in narrow spaces.

What we have done multiple times is have the rental car delivered, Europcar does this for no extra fee (or at least they did pre-covid).

What we usually do is arrange for a private transfer (sprinter van or sedan) to greet us at the train station or airport and drive us to our accommodations. We then have the car delivered to where we are staying. The last time we did this was when we had a villa outside of Paciano in Umbria. They brought the rental car to us, we did the paperwork sitting outside on the veranda, and when we left, we just left the keys in the villa, and they picked it up for us. It was so convenient. Other times they’ve left the car and paperwork with the hotel where we were staying.

We don’t drive much when we are in Italy, I find it a little stressful, but when you are at a villa or an agriturismo outside a city, you really need it, if only to make runs to the market. For full day sightseeing trips, we’ve found it to be much more convenient to book a driver and car for the day. It is more expensive, but not having to worry about getting lost, or finding parking, or driving through a traffic controlled area (of which there are many, and you get the ticket in the mail after the fact) makes it a worthwhile splurge.

Enjoy Lucca, it is such a cute town, one of our favorite places to spend a few days, and it is not as overrun as San Gimignano or Volterra. Riding a rental bike around the ramparts is a fun afternoon, there are places to stop and have a espresso or some wine, and it’s just a great two hour “live like a local” experience.