No. There are very very very few exceptions to HIPPA. Most of them are Law Enforcement or Regulators whose jurisdiction covers public health (such as CDC or OSHA) but they also have to maintain your privacy.

An employer/organization/friend/colleague/whatever do not usually have the legal jurisdiction to check up on your health status with your medical provider. The only exception to this is if a specific regulation requires your employer to know such as required Hepatitis B vaccination, which an employer is required to provide if you're reasonably expected to come into contact with bodily fluids. But again, this is very specific.

So in short, they can ask, but your medical provider, and I say this with certainty, will not provide that information to them.