Maybe, probably not. The leading case would appear to be Raines v. Byrd, 521 US 811, where 6 congressmen sued over a line-item veto law (later held to be unconstitutional). The court notes the established legal fact that
To meet the standing requirements of Article III, "[a] plaintiff must
allege personal injury fairly traceable to the defendant's allegedly
unlawful conduct and likely to be redressed by the requested relief".
where that court added the italics; and
the alleged injury must be legally and judicially cognizable. This
requires, among other things, that the plaintiff have suffered "an
invasion of a legally protected interest which is . . . concrete and
In this case, the court find that "appellees have not been singled out for specially unfavorable treatment as opposed to other Members of their respective bodies", and they are claiming institutional injury, arising "solely because they are Members of Congress...If one of the Members were to retire tomorrow, he would no longer have a claim; the claim would be possessed by his successor instead". The court did, however, uphold standing in one legislative case, Coleman v. Miller, 307 U.S. 433, where there was an issue over whether the legislature had ratified a constitutional amendment when the Lt. Governor of Kansas cast a tie-breaking vote on the question (the allegation was that this was improper). The court held that the legislators "have a plain, direct and adequate interest in maintaining the effectiveness of their votes", finding that their
votes against ratification have been overridden and virtually held for
naught although if they are right in their contentions their votes
would have been sufficient to de- feat ratification. We think that
these senators have a plain, direct and adequate interest in
maintaining the effectiveness of their votes.
The Raines court finds that legislators "have standing to sue if that legislative action goes into effect (or does not go into effect), on the ground that their votes have been completely nullified" – but that was not the case in Raines (the line-item veto act had clearly passed).
Applied to the Oprah hypothetical, senatorial vote would arguably have been completely nullified by the proposed process: that is the argument made in the present complaint, para 33-34. The end of sect. III of the Raines opinion gives extensive historical analysis of the fact that branches of government do not generally have standing to sue each other, closing with the note that
Our regime contemplates a more restricted role for Article III courts,
well expressed by Justice Powell in his concurring opinion in United
States v. Richardson, 418 U. S. 166 (1974):
"The irreplaceable value of the power articulated by Mr. Chief Justice
Marshall [in Marbury v. Madison, 1 Cranch 137 (1803),] lies in the
protection it has afforded the constitutional rights and liberties of
individual citizens and minority groups against oppressive or
discriminatory government action. It is this role, not some amorphous
general supervision of the operations of government, that has
maintained public esteem for the federal courts and has permitted the
peaceful coexistence of the countermajoritarian implications of
judicial review and the democratic principles upon which our Federal
Government in the final analysis rests.
There is a very narrow window through which the Senate might have standing to sue POTUS and otherwise, the answer is "no". The Oprah case is distinguished from Raines in that there is no political recourse to simply ignoring the appointments clause, except impeachment, and the courts might see such an action as equivalent to vote-nullification.