These organizations are, by definition NON-GOVERNMENT organizations, and, as such, are under no requirements to meet any kind of special loyalty test.
However, I'm also not clear that accepting donations from Russian-related groups for non-political purposes somehow demonstrates disloyalty, either.
If the groups are non-profit, they are not allowed to give to political organizations. These groups and the advocacy groups are required to create "sister" organizations for reporting their any kind of direct advocacy, political donations or lobbying efforts. Donations to, say, the Planned Parenthood arm that does political lobbying, for instance, is not considered a charitable/tax-deductible contribution. The charitable, non-profit Planned Parenthood organization, in turn, would not be able to legally send money to the "PoloHoleSet Fight The K-Dog" political lobbying organization, though their separate political arm would.
As such, since it is illegal and these organizations have to report their finances, it's doubtful that they have been making political contributions.
If it's shown that they are assisting a group's illegal activities, if ties to certain groups are banned by law, or if it's shown they are disbursing funds in a way contrary to the way I outlined above, then, yes, there would be repercussions. I don't think you've offered any kind of evidence that this is happening, though.
If they only happen to be working with, are supported by or are supporting a group you don't like very much, that's very much protected by the First Amendment, and you couldn't ban that.