Do we have to purchase our train tickets from Poggibonsi to Florence in advance? Or...can we just show up at the train station in JUNE and expect to get on the train of our choice?

No need for advance purchase. Just show up and purchase the tickets of your choice to Poggibonsi. The train is often enough that I would not be concerned, however, if you are, just purchase the ticket of your choice the day before your planned train trip.

You can check schedules on www.trenitalia.com

I've never heard of anyone buying advance tickets for a 25 mile ride on what I assume is a commuter train....

And the bus goes every 30 minutes I believe.

tedgale..
on the day we want to visit Florence the trains leave at 8:35am, 9:13am and then not until 11:24. I am making reservations for the museums so, if we miss the departure time we need we will miss our reservations in Florence.

As no-one can absolutely guarantee that unreserved tickets will be available and as you MUST get to Florence, you really have no option but to reserve....however remote the possibility of a sold-out train.....

I have we've answered your question clearly:

No, you don't have to purchase your train tickets from Poggibonsi to Florence in advance.

Yes, you can just show up at the train station in JUNE and expect to get on the train of your choice.

As a matter of fact, on these regional trains, I don't think there's seat reservations. So, I don't see any advantage in buying the tickets in advance. But like others said, if you go by a train station in Italy in the days prior to day of travel, then just get it then.