Wait, what? You're dedicating 1 day to Florence and Pisa? That is just not enough time to see all you need to see in each place. But not knowing your plans before and after.... I'll comment more later ;-).
Aside from that, yes, you can deposit luggage in Pisa Centrale (not sure about La Spezia) and pay per hour it is kept. If you're making your way up to the Cinque Terre and stay a night in Vernazza, where are you headed afterwards? If back to Tuscany, it would make sense to leave part of your luggage behind but it does force you to go back through Pisa to get pick it up.
Otherwise, I would take my luggage with me all the way to Vernazza, check in at your place for the night so leaving your luggage there and then head out to hike with only what you need for the day hike. This way you are also not carrying all day what you need for the overnight stay. We did the hike from the north and got to Corniglia, got soaked in a spring afternoon downpour, our B&B was in Manarola so when we go there to check in most of our stuff was humid at best. We did the last town, Riomaggiore, that evening as we headed to dinner there ;-) So it is doable in the opposite direction.
Aside from weather, you will also have to check the state of the trails by October - they get cleaned up as best as they can for the high season (which October is not) but by October if there is rainfall and landslides, they might be closed. So maintain a high degree of flexibility as far as hiking at CT goes. There is always the train that connects them.
Another option: if you're staying in Florence before and heading back through Florence after your stay in Vernazza, why not see if your hotel or B&B will keep your luggage for the time you are away? They often have closets where they keep luggage for guests who check out in the morning but are planning to be in Florence for a few more hours.... so it is something you can also look into.