... are simply polling people.

Every state is recording Covid-19 vaccination information, except for New Hampshire for whatever reason. (They might record immunizations in some other way, but I'm not sure.)

When there's a coronavirus vaccine, how will we make sure everybody gets it? That's the job of state immunization registries.
https://www.usatoday.com/story/news/health/2020/06/12/coronavirus-vaccine-state-immunization-registries/5321892002/

Ohio, as just one example.
https://www.ohiopublichealthreporting.info/Enrollment/FileSystem/HL7/Public_Instructions_to_access_ImpactSIIS_record.pdf
"COVID-19 vaccine providers in Ohio are required to report patient information into the ImpactSIIS system
when the COVID-19 vaccine is administered. If you have received the COVID-19 vaccine in Ohio, your
information should be in the system."


Hence the redder, Trump-supporting states are definitely lagging behind with vaccinations so far, in general.