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That will depend on the nature of their funding. In most cases, government funding does NOT covert the employer into a government employer. They are contractor or independent private organization, which would then be subject to the NLRB because that entity has jurisdiction over any employer that is not a government employer.
I am not personally aware of any non-profits that have taken on a government agency mantel, but I suppose that it could happen where a non-profit is coopted by a government agency and ultimately just becomes another arm of the government.
In every circumstance though that I have seen with funded non-profits, they maintain their private entity status and would subject to the NLRB.