The Texas Department of Emergency Management has set up an on-line portal showing the availability of COVID 19 Vaccine in every county in Texas. It lists both hospitals and pharmacies and the number of vaccines on hand. The site is updated daily.
Find the info here: https://tdem.maps.arcgis.com/apps/webappviewer/index.html?id=3700a84845c5470cb0dc3ddace5c376b