It is unusual, though not unheard of, for charities to impose non-compete provisions in employment contracts with their employees.  I am not aware of any governmental funding programs that would make such provisions illegal.  If that is the case, the legality is a function of normal employment law.

In order for a non-compete to be effective, it has to be based on legal consideration on both sides.  An employer could not unilaterally impose such a condition upon an employee under contract for work.  It could impose the provision in a renewal of the contract so long as the employee has the opportunity to decline and look for work elsewhere.  If the employee accepts the new term, it would be effective, assuming it is sufficiently limited in duration, scope, and area.

For an at-will employee who effectively comes to work every day without a contract, the employer could impose the non-compete simply by notice of the change in terms of the relationship, and it would be effective if the worker continued to come to work.  But if employees were never told of the change, and did not know about it until they sought employment elsewhere, they might have an argument that it was ineffective because they never accepted what they never heard of.  It would probably be a tough argument because the employer would undoubtedly argue that “everybody knew” of it because it was printed in some publication or other.  But there might be a basis for negotiation.

UPDATE: 5/2024

The recently approved Final Rule of the Federal Trade Commission prohibiting non-compete agreements in most employment situations applies to corporations only if they are "organized to carry on business for its own profit or that of its members."  But in the notice announcing the Rule, the Commission recites precedent applying FTC rules to nonprofit entities "if they are in fact profit-making enterprises."  Most of the exceptions to the exemption appear to be in medical situations such as hospitals or physicians' groups, but the exact contour of the rule will have to be decided by later decisions.