The House, in turn, is preparing to file a lawsuit as early as next week to try to get the federal courts to strike down the administration’s immunity theory, which has been advanced by presidents of both parties but has never been fully tested in court. Even though the suit will probably take aim at Ms. Donaldson’s former boss, the onetime White House counsel Donald F. McGahn II, a ruling in the Democrats’ favor could effectively force Ms. Donaldson and other witnesses who have defied the Judiciary Committee to take the witness stand.

That court fight could take months, but without full access to relevant witnesses, Democratic efforts to investigate and air possible obstruction and other misdeeds documented by Mr. Mueller have all but ground to a halt.

Democrats are struggling to gain traction on other investigations as well. On Monday, Representative Elijah E. Cummings of Maryland, the chairman of the House Oversight and Reform Committee, wrote a follow-up letter to Mick Mulvaney, the acting White House chief of staff, demanding answers about what happened to records of Mr. Trump’s meetings with President Vladimir V. Putin of Russia after reports that Mr. Trump tried to conceal them.

Ms. Donaldson, as chief of staff to Mr. McGahn, witnessed or was privy to some of the most explosive moments detailed by Mr. Mueller’s investigators, including the firing of James B. Comey as F.B.I. director and attempts by Mr. Trump to gain control of the investigation. Ms. Donaldson kept detailed notes of those episodes and others, which are referenced frequently in Mr. Mueller’s 448-page report.

“Ms. Donaldson had a front-row seat to many of the instances outlined in the Mueller report dealing with President Trump’s alleged obstruction of justice and other abuses of power,” Representative Jerrold Nadler of New York, the Judiciary Committee’s chairman, said in a statement on Monday, “which is why she is a key witness for the committee in our ongoing work to hold the president, his associates and members of his administration accountable.”