About 250 civil rights activists marched across the 14th Street bridge, down Pennsylvania Avenue and up the steps of the U.S. Capitol yesterday, culminating a 64-day "Pilgrimage to Washington" designed to churn support for passage of the 1982 Voting Rights Act.

Also yesterday, a group of poor people, coordinated by the Association of Community Organizations for Reform Now, set up a "tent city" behind the White House as a protest against President Reagan's housing policies.

The march, led by about 60 Southern Christian Leaderhip Conference "pilgrims" who left Tuskegee, Ala., in April on a five-state voter-registration tour, arrived on the Capitol steps just as the House of Representatives voted unanimously to approve the act.

The timing was hailed as providential. "God has brought us here at a very peculiar moment," said the Rev. Joseph E. Lowery, president of the SCLC, to applause and shouts of "Hallelujah."

"Now it's time to take that bipartisan sensitivity in Congress and address other minority concerns--unemployment, housing and welfare," Lowery said.

The protesters, most of them young and spirited, criticized the policies of the Reagan administration and attacked political apathy among blacks with chants and songs: "Tell Ronald Reagan time is winding up / So much corruption in the land / Why don't you people take a stand / Time, oh, time is winding up."

In their trek through the South, the SCLC marchers had conducted door-to-door canvassing for voter registration in 30 congressional districts judged particularly vital to black voters. Lowery said the procession was designed, in part, to stir up memories of the civil rights struggles of the 1960s--to remind blacks that "The Struggle" is not yet over.

To emphasize that point, the demonstrators plan to erect a symbolic "Resurrection City II" on the Ellipse, harking back to the 1968 Resurrection City encampment on the Mall, where thousands set up temporary residence under shelter of plywood and plastic in an effort to draw attention to the plight of the nation's poor.

The day's march--evocative of those earlier days, but also modest enough to illustrate how times have changed--began at 10 a.m. in brilliant sunshine and mild temperatures at St. John's Baptist Church on Columbia Pike in Arlington.

The group's numbers swelled along the way as supporters gradually arrived. The marchers chanted and sang along the way, drawing a limited response from bystanders.

As the crowd reached the Washington side of the 14th Street bridge, a gusty wind ceased momentarily and the demonstrators, walking in rows, stopped to kneel and pray.

"This bridge is symbolic of the bridges of unemployment, housing and opposition that we must cross," said Spiver Gordon, 42, vice president of the SCLC's Alabama branch. "We realize that what's ahead is hard times and continued struggle. And we need more black people to get involved."

The Rev. Spencer Coleman, 70, of Alexandria, wore a red SCLC baseball cap as he crossed the bridge and talked about how "it's getting worse for blacks. At my age, I can see it."

Coleman, who helped integrate Alexandria schools years ago, said, "I march as a minister. Moses marched. The children of Israel . . . Sojourner Truth marched. The benefit of our marching is that we let the world know that we are not satisfied with the way the heads of state are operating."

The marchers said they have received donations and support from churches, labor unions, city leaders and numerous individuals. Marching with the SCLC yesterday was what looked like a version of the old liberals-and-labor coalition that was so often in evidence during the headiest days of the civil rights movement: whites and American Indians, the American Federation of Government Employees, the Southern Rural Women's Network.

Joining them on the steps of the Capitol yesterday were House Majority Leader James C. Wright Jr. (D-Tex.), Rep. Parren Mitchell (D-Md.) and D.C. Del. Walter Fauntroy, as well as Coretta Scott King, widow of the Rev. Martin Luther King Jr., and D.C. City Council member Wilhelmina J. Rolark (D-Ward 8).

ACORN's three-day protest seeks to persuade housing authorities to turn over tax-delinquent properties to the poor for renovation, according to spokesman Clarence Anderson of Boston. The demonstrators are from about a dozen cities.

Yesterday, some of the group, chanting and waving signs, surged to the gates of the White House, but none attempted to enter the grounds and there were no arrests.