We will be staying in Cortona next summer for six weeks and wanted to know if there are residents who open their home for local dinner or lunch - we were looking to experience local foods in or around Cortona
One website is https://www.eatwith.com/ but it didn't seem to have much in Italy. There is a home dining organization here, though none of the Italian listings are close to Cortona: http://homehostedmeals.com/find-host.html
One reference source for locally oriented food is the Slow Food guide to restaurants, Osterie d'Italia. To be listed, a restaurant must offer traditional foods with reasonable prices usually in a traditional ambiance. The ones we've eaten in are not tourist oriented or at least not foreign tourist oriented. The English version is on Amazon.com, but it's from 2007. The Italian version is 2018 on Amazon.it.
Cortona has a very helpful TI office on the main street. You could try to email them or simply stop and inquire once you arrive. Rough Guide or Lonely Planet guidebooks on Italy or Tuscany might have an email address for the Cortona TI office.
One of the best known home-hosted meal organizations in Italy is Cesarine.it. They are affiliated with Slow Food. The have several offerings in Cortona. That’s where I would start.