The legal challenges, even if they are unlikely to succeed in court, will give Mueller some pause before he takes any major actions, said Joe Moreno, a former federal prosecutor who is now a partner in Cadwalader's white collar defense and investigations group.
"Mueller will not want anything he does second-guessed because [Whitaker's] authority may be questioned," Moreno said in an email.
It is not clear when the legal challenges to Whitaker will be resolved.
Tom Goldstein, a prominent Washington lawyer, urged the Supreme Court on Friday to take up the issue of Whitaker's appointment immediately, citing the difficulty of "unwinding" all of the decisions that Whitaker makes while in his role. Those decisions would include approving subpoenas or indictments, or even firing the special counsel, Goldstein told CNBC.
The other challenges could take longer. In a case brought by the state of Maryland, which is also represented by Goldstein, a district court in the state has scheduled arguments for late 2019, and has not indicated when it plans to make a ruling. Goldstein has sought a preliminary injunction in the case that would bar Whitaker from carrying out his official duties.
Attorneys representing a client in an immigration case before the 2nd Circuit Court of Appeals have filed for a similar preliminary injunction. The government has until Dec. 10 to respond to their claim.
On Monday, Senate Democrats Mazie Hirono, Richard Blumenthal and Sheldon Whitehouse brought their own case challenging the constitutionality of Whitaker's appointment. That case, the latest, is likely to run into technical hurdles about standing and possible remedies, legal experts say.
Critics have called on Whitaker to resign from his oversight role, citing his previous criticism of Mueller's investigation and his close ties to Sam Clovis, a witness in the investigation. Whitaker has said there was "no collusion" between the Trump team and Russia, an echo of Trump and his attorneys' own language.
Mueller's prosecutors broached the topic of Whitaker's effect on the probe in a court filing Monday. A panel of judges in Washington is overseeing a case brought by a target of the probe, Andrew Miller, who was held in contempt after failing to comply with grand jury subpoenas obtained by Mueller in May and June.
The special counsel's office wrote in its brief that Whitaker's appointment had "no effect" on the grand jury subpoenas issued to Miller. But, they wrote, that was because the subpoenas were issued before Whitaker was appointed. Any "new" challenges should be argued in a different venue, they wrote.
A spokesman for the special counsel declined to comment. The White House did not immediately respond to a request for comment.
To be sure, there are ways that Mueller can go about inoculating himself from challenges to future actions. For instance, Mueller could obtain approvals from both Whitaker and Rosenstein. But it may not be politically viable for Whitaker to do so if it is perceived to be in response to questions about his legitimacy.
In a statement, the Justice Department said Whitaker's appointment "is lawful and comports with the Federal Vacancies Reform Act, the Appointments Clause of the U.S. Constitution, Supreme Court precedent, past Department of Justice opinions, and actions of U.S. Presidents, both Republican and Democrat."
The Office of Legal Counsel signed off on Whitaker's appointment in a 20-page legal memo earlier this month.
Mueller's use of a grand jury could also bolster his legal case against any future challenges.
Willy Jay, a former assistant to the solicitor general and a partner at the law firm Goodwin Procter, said that "there are an awful lot of things that generally cannot be litigated after an indictment."
"The grand jury returns the indictment, it is not the attorney general," Jay said. "Once the grand jury returns the indictment, things that go into that cannot be second guessed."
Jay also pointed to the de facto officer doctrine, under which actions taken by officials can be upheld even if it is later discovered that they were improperly appointed. But that would not necessarily apply if the target of a subpoena or indictment challenges Whitaker's appointment at their first opportunity to do so.
A case from 2016 involving actions taken by the head of the Consumer Financial Protection Bureau could be instructive.
The case arose after the Supreme Court ruled that Richard Cordray, the first head of the CFPB, was improperly appointed by President Barack Obama in 2012. Obama reappointed Cordray in 2013, after which Cordray retroactively affirmed the actions he had taken while he had been improperly serving.
A federal appeals court in California upheld decisions Cordray had made before his renomination. The Supreme Court, however, has never said that the courts can retroactively bless the decisions of an official appointed unconstitutionally, Jay said.