What makes you think that “By law, I guess they would - it’s a big enough organization.” In the U.S., there is NO law compelling any organization, large or small, to provide health insurance. So they don’t have to if they don’t want to.
If an organization does choose to provide health insurance to employees (and most large organizations do), then there are laws about what that program can and cannot do to be eligible for favorable tax treatment. However, there is no requirement to do anything.
I presume that they are NOT exempt from paying Medicare premiums.