On New Year's Eve, George sent out a newsletter to members of the community to inform them on where the county stands in regard to vaccinations.
As of now, healthcare workers are administering doses around the clock to frontline workers, those above the age of 65 and others who are in a priority group assigned by the state.
SEE ALSO: Breakdown of when and where to get COVID-19 vaccine
For other residents hoping to receive treatment, George said state officials could send another round of vaccines to the area as early as tomorrow. But, as of now, there is still no clear arrival date.
"Unfortunately, like many local governments in similar situations, we have no idea when the State of Texas will ship more or how many, which is why we are prepared for all situations," George said.
SEE ALSO: Top Houston-area doctor gets blunt about slow COVID-19 vaccine rollout
What's in store for 2021? Health experts say year looks promising
He continued the newsletter by pointing out the lag in distribution across the state, despite efforts from Pfizer and Moderna to follow "Operation Warp Speed".
"I have asked the state for more vaccines for Fort Bend, so we can get it out at hyper-speed to all that need and want them - with the goal of making it universally accessible," he said.
Many county leaders continue to ask Gov. Abbott to open a new phase of distribution that would vaccinate those in professions such as teachers, police officers and other critical frontline staff who can't work from home.
Visit the Texas Department of State Health Services website for a list of providers distributing the vaccine.