I look at this website every week to look at the most recent vaccine allocation chart (links on the right side of page):
https://www.dshs.texas.gov/coronavirus/immunize/vaccine.aspxThe pharmacies, medical centers, etc show up here before they are shown on the map.
I saw that my local Tom Thumb pharmacy was allocated 100 doses this week, so my wife called and asked about getting on a list (as of today they still haven't received the doses). They told her to go to their website and sign up for their updates. So, she did. Not sure what that does other than letting her know when doses are available.
Point is, you might try looking at the allocation charts and see who is allocated doses near you, then contact them.