Is the federal government collecting vaccine data to keep tabs on which American citizens are following orders?
The federal government does not keep records on which American citizens have been vaccinated.
According to the Centers for Disease Control, "there is no national organization that maintains vaccination records." The only records that exist are those in local vaccination clinics and the paperwork patients receive after getting vaccinated.
The Atlantic reported that the U.S. operates on an "honor system" — there is often no way to validate whether vaccination cards required for entry into certain places are legitimate.
The federal government has also never legally ordered the general public to get vaccinated. While more limited mandates have been enacted for public schoolchildren, federal employees, etc., there are exemptions that allow the immunocompromised or those with "religious or philosophical objections" to opt out. Other alternatives include periodically showing a negative test result or paying a fine.