What will be open/closed May 1?

Stores will be largely closed, except for some chain stores, which may open at some locations but with limited hours.
I would plan to do your grocery shopping the day before, especially in a small town, where stores are often small independently owned grocery shops.

If you tell us which small town you are referring to, I can try to see if there are any openings available. Things change from locality to locality, and some areas visited by large number of tourists may have more options.

Thank you, Roberto,
On May 1 we are driving from Tivoli to Montepulciano.
Do you think the Conad in Montepulciano (at the bottom of the hill) will be open?

Conad has a search engine for stores open on May 1st in its chain:
https://www.conad.it/news-e-approfondimenti/supermercati-conad-aperti-primo-maggio

It looks that no one in the Siena province is open. If you look at timetables of the Montepulciano one on google, it looks closed also on May 2nd.

The largest chain of grocery stores in that area of Tuscany is actually Coop. They are larger and they are in Pienza, in Chiusi, in Chianciano, in Acquaviva, in Sinalunga. However in the last couple of years Coop has always stated that in honor of their workers and their families, they will observe the International Workers’ Day holiday nationwide, except for very selected Hyper-Coop (the chain’s humongous superstores), of which there are none near Montepulciano.

Your best bet is to stop at an Autogrill or a Chef Express at any A1 freeway rest area (Area di Servizio). Inside all of them there is a restaurant, a deli, a coffee bar and also a sort of a mini market, that way you can eat something at the restaurant or even buy some selected groceries. They are generally open all the time, including on May 1st. I’m also sure that restaurants should be open in Montepulciano and maybe some tourist shops. For sure you should be able to buy some Pecorino di Pienza and some bottles of Brunello or Rosso di Montepulciano to drink. Those shops cater to tourists so they should be open on May 1st.

I don't think you will find grocery stores open in smaller towns. I usually travel for a few days then and have not had any grocery stores open in small towns. As mentioned COOP announces closures for both 25 April and 1 May - they put out a big sign every year.

Thank you all for your kind and generous replies. We will plan on stocking up on April 30.
Looking forward to our trip!