Anne Peterson leaned wearily on her crutch yesterday noon and hugged the shade of First Rising Mount Zion Baptist Church at Sixth and N streets NW, near the head of a thick line that snaked around the church and halfway down the long block.

"I've been here since 7:30 this morning," she said. "They said they were giving out chicken and I want to get some. But I need to get home--my son's been unemployed and he has to go see about a job."

It was nearly 12:30 p.m. when the church doors finally opened and volunteers dropped into Peterson's plastic shopping bag, not a chicken, but a five-pound box of macaroni, several sweet potatoes and a bag of powdered eggs.

Peterson was one of an estimated 15,000 persons--all certifiably needy--to benefit from the 34th food distribution of the Council of Churches of Greater Washington. Since the program began 19 months ago, the number of people who turn out for the free food at 16 churches throughout the city has grown steadily from 2,600 persons to 15,000, said the Rev. Rodney Young, who manages the program for the council.

Despite the talk of economic upturn, said Young, "There are still large numbers of unemployed; the social service programs have still been cut. It would amaze you, the number of college graduates in that line. They are there because they don't have a job."

Most of those in the line at First Rising Mount Zion, one of the largest distribution centers, were, like Peterson, elderly, about evenly divided between men and women. Some brought children, who alternately fretted, played or slept through the long hot wait.

The church council's program began in January 1982 by distributing surplus cheese, and later butter, from the Department of Agriculture. Earlier this year the Virginia Conference of the United Methodist Church supplemented the surplus commodities with produce grown nearby. The sweet potatoes handed out yesterday were part of that effort, as were 40,000 pounds of white potatoes distributed earlier in the summer, Young said.

The Methodists have also pledged a supply of fresh green beans, cucumbers and fresh greens, as they mature on Eastern Shore farms where they were planted for the program.

The number of surplus government commodities--the backbone of the program--also is expanding to include honey, cornmeal, frozen chicken and powdered milk and eggs.

The Council of Churches enforces firm guidelines as to who can receive the free food, insisting the recipients be on food stamps, Medicare, public assistance or Social Security; those on Aid to Families with Dependent Children and the unemployed are also eligible.

The free food makes a difference to people like Peterson who, with five mouths to feed, says she lives close to the edge. "I get $387 a month disability, but my rent has gone up to $260," she said. "My son said he'd help me when he gets a job," she added hopefully.

There is no fixed schedule for the distributions. Sometimes Young knows a week or 10 days ahead that commodities will be available; sometimes he has only a couple of days to organize getting trucks or trailers to First Rising Mount Zion, to mobilize the transportation and volunteers from the 15 other churches in the program to unload and divide the food and gear up their distribution systems.

The needy find out about the distributions from notices posted on church doors, the media, or more often, from the neighborhood grapevine.

Although there were a few incidents of line crashing and shoving yesterday as people waited long hours in the heat, Young said security has not been a problem.

As Young said happens increasingly, the supply of food at First Rising Mount Zion gave out yesterday while there were 150 or 200 people still in line. "They were disappointed. . . . I didn"t sense any anger," he said. "They've been there a number of times and this time they didn't get served."