No,
employees of non-profits, as a general rule,
are not exempt from federal income tax. Sorry.
When she filed the W-4 with her company, she might have indicated on the W-4 that she swore that she was not liable for federal withholding. It's rare, but it happens.
However, if your spouse is like most employees of a non-profit, she will owe federal income tax, and the company should have been withholding at a rate consistent with the information she gave on the W-4.
Yes, she needs to talk to the company. There may be other, rare circumstances that the company is aware of.
