I would but I only fight battles worth winning. Quite simply, almost 100 years of legal precedents mean you're basically pissing into the wind by not receiving the COVID vaccine when the POTUS signed the Executive Order mandating federal employees and contractors receive the COVID vaccine.
At best, you are only entitled to accommodation or exclusion if you 1) have a “sincerely held religious belief” (see Title VII of the Civil Rights Act of 1964), 2) notify the FAA you have a preexisting disability that prevents you from safely receiving a COVID vaccination (see Rehabilitation Act of 1973), or 3) you have a medical condition that may worsen upon taking the COVID vaccine (see Americans with Disabilities Act of 1990). And since you would have had to declare any preexisting disabilities or medical conditions (#2 and #3 above) to the FAA as a condition of employment when you were hired, you're really only left with an accommodation under #1 above. Should you attempt to claim via preexisting disability or medical condition, and you did not previously advise the FAA, you set yourself up for termination for lying to the Flight Surgeon when you submitted your medical paperwork.
While the COVID vaccine mandate applies to federal employees and contractors, the DOT/FAA may grant reasonable accommodation requests to those with valid medical or religious reasons. However, an accommodation is normally only granted if it does not cause an undue hardship on the employer - not employee. Undue hardship means requiring significant difficulty or expense on the part of the employer. Furthermore, an accommodation to an unvaccinated employee must not pose a “direct threat.” An employee creates a direct threat if, even with an accommodation, they create "a significant risk of substantial harm to the health or safety of the employee or others that cannot be eliminated or reduced by reasonable accommodation." A requirement to work side-by-side and in close proximity to co-workers - especially when training - creates that "significant risk of substantial harm to...others that cannot be eliminated or reduced by reasonable accommodation."
One final nail in the coffin is that it appears NATCA has chosen not to fight the Executive Order as a change in working conditions... and they are unlikely to do so. Since unions represent members, as opposed to defend members, the law (and legal precedents) means you're out of luck on any failure to represent claim against the union.
It is my personal belief that few reasonble accommodation requests will be granted. And if you served in the military and received vaccinations as a requirement to service, you're definitely going to be denied a reasonable accommodation no matter how sincere your religious beliefs. And if you did not serve in the military, but received flu vaccinations at any time, you're probably going to be denied a reasonable accommodation no matter how sincere your religious beliefs. So, if you want to remain an air traffic controller, you can either get vaccinated or get punished/terminated.
