The twin evils of government fraud and waste loomed large on President Reagan's campaign enemies list and yesterday the president formed a new organization with a squeaky-clean name, the President's Council on Integrity and Efficiency, to root the two from federal programs.

At the same time, the president appointed the first six inspectors general to fill the 16 vacancies he created when he fired all of these agency watchdogs on his first full day in office Jan. 21. Five of the six Reagan chose yesterday were among the 16 he fired.

"We will not rest with today's announcement. We will not simply tuck this event away and go on with business as usual," the president said as he signed the executive order establishing the new council.

"We are going to follow every lead, root out every incompetent, and prosecute any crook we find who's cheating the people of this nation," Reagan added.

The anti-fraud and -waste council will be chaired by yedwin Harper, deputy director of the Office of Management and Budget, and is designed to facilitate interdepartmental cooperation, the president said.

Harper said that while no additional staff will be given to the inspectors general, they will have greater resources because of closer cooperation with the FBI.

He said that the audit functions of the government are estimated to have saved $4 billion over the last two years. "That's a fairly hard figure," he said.

In introducing five of the six newly appointed inspectors general to reporters, Harper insisted that it had made sense to fire them all and then examine their qualifications. He described the inspectors' functions as vital but denied that firing all of them had disrupted the work of their offices.

Not all was running smoothly even with the appointment process, however. Frank S. Sato, who before Jan. 21 was the inspector general at the Transportation Department and was appointed to the same position at the Environmental Protection Agency yesterday, was absent, apparently because he had no advance warning of his new appointment. One of the other inspectors said Sato didn't know of his new job as of yesterday morning.

Harper said the 10 other inspectors general will be named very soon and that two of them will be persons who held such jobs before the Reagan purge.

The purge has been criticized by various members of Congress, mostly Democrats, who have said that the supposedly nonpartisan inspectors were being politicized by the Reagan White House. In the latest such charge, six House Commerce and Energy subcommittee chairmen sent a protest to the White house charging that White House political director Lyn Nofziger and other White House aides were playing significant roles in choosing the new inspectors.

Harper denied the charge. He said Nofziger saw the final list of candidates, but "he did not veto any candidate."

In addition to Sato, the inspector general named yesterday are: Robert L. Brown, a retired Foreign Service officer, for the State Department; Paul R. Boucher, reappointed to the Small Business Administration; James B. ythomas, reappointed to the Education Department; Thomas F. McBride, moved from the Agriculture Department to the Labor Department, and Charles L. Dempsey, reappointed to the Department of Housing and Urban Development.