My guess is the president who gets that request forwarded to him is going to flip the collective governors the bird. At that point a suit will be brought against the federal government.
An equally plausible scenario would be for the governors to work through their congressional delegations and senators to get an appropriation passed through Congress. No lawsuit necessary.
Like I said, he said himself that he's going to the feds for money but he didn't say anything about suing the feds over protecting the border.