Advocate for the homeless Mitch Snyder, charging that federal officials are not fulfilling President Reagan's promise last fall to renovate a homeless shelter near Capitol Hill, delivered a letter of protest to the White House yesterday and asked for a meeting with Reagan to discuss his concerns.

"Your promise to us, to the residents of this shelter, and to the homeless of America has been broken," Snyder said in the letter addressed to Reagan.

Reagan approved an agreement last fall to renovate the shelter, ending a dramatic and well-publicized 51-day hunger strike by Snyder, but since then Snyder's estimate on how much a proper rehabilitation will cost has doubled from $5 million to $10 million.

The current impasse between Snyder and federal officials is only the latest in a cyclical love-hate relationship that has developed between the radical advocate for the homeless and the conservative Reagan administration over the shelter at Second and D streets NW.

What started as a plea by Snyder for temporary use of the run-down, federally owned building as a cold-weather shelter two winters ago has become a demand for $10 million in renovations to make it into a model shelter for the homeless.

Harvey Vieth, chairman of the federal Task Force on Food and Shelter, said yesterday that the General Services Administration is ready to start the renovation any time and that the federal government plans to spend up to $5 million as promised to make necessary repairs.

"We are absolutely living up to our commitment, and I don't know what Mitch's problem is," Vieth said. "He wants twice the $5 million now, and so I think there is a problem."

At the White House, deputy press secretary Larry Speakes said the administration has been "very responsive to the needs of the homeless in Washington, and we particularly worked with Mitch Snyder." He said he did not know if the president had received the letter.

Snyder said federal officials have refused to develop any detailed specifications for the work to be done and he charged that they were trying to get away with doing a "patch-up" job.

Vieth said GSA wants to save time by designing as the work proceeds. He said GSA and Snyder are not that far apart on their ideas of what should be done but that Snyder's design calls for such things as individual cubicles that GSA does not intend to provide.

Vieth said the rehabilitation needs to last for only five to 10 years, because the shelter is not intended to be a permanent institution.

Snyder originally got the federal government to agree to turn over the building, which used to be part of the Federal City College campus, by promising to take responsibility for running and maintaining it during the winter months.

When the 800-bed shelter opened in January 1984, Health and Human Services Secretary Margaret M. Heckler was on hand for a celebration of the joint effort between the Reagan administration and Snyder's group, the Community for Creative Non-Violence.

But by spring, the CCNV protested the Reagan administration's plan to go ahead with an auction of the building, demanding that it stay open in the warm months as well, and Reagan ordered a delay of the sale. Snyder started a hunger strike in September that resulted in the president's agreement to the renovation, and he and Vieth since have been trying to negotiate how to accomplish it.

Snyder said if the president does not order officials to make the renovations the CCNV wants, the group will go to federal court to try to force the federal government to make emergency repairs and to staff the facility.