That's just a backdoor for giving more power to the federal branch. Its like 'interstate commerce' where anything that has interstate implications (pretty much everything) can be influenced by the federal government. What if an anti-abortion president elected someone to this board and told them that no health care provider that accepts medicare or medicaid can offer abortions?
I don't see mandating people to get a vaccine that they don't want as non-political bureaucratic action, especially considering its coming from the president's office. It's mandating a medical treatment. Take a step back and ask under what authority and supervision should we require a government to be mandating a medical treatment.