Not sure if it will speed things up, but after clicking on that "basic" site for MyChart I mentioned above, and entering "Texas", got a list of numerous Health care providers/networks.
I looked and found that you still must have an appointment to get the vaccine, and only the County health department can issue those appointments.
"All you've got to do is own up to your ignorance honestly, and you'll find people who are eager to fill your head with information" --Walt Disney
