In a quiet chapel in a corner of the Washington Cathedral, under a vaulted stone ceiling and a stained glass window depicting the Crucifixion, volunteers yesterday began reading aloud the names of 57,939 Americans killed or missing in the war in Vietnam.

The volunteers stood beside a wooden altar set with two candles, white chrysanthemums and bamboo. Speaking in voices that sometimes barely rose above a whisper, they alternated in reading the names alphabetically from computer lists, beginning a somber memorial vigil that is to continue round-the-clock until midnight Friday.

Through the day, several hundred persons from around the country -- friends, family members and military veterans -- filed in and out of the cathedral's War Memorial Chapel. Some heard the name they had come to hear and wept openly. The eyes of men in uniform filled with tears. Occasionally the ceremony faltered as readers, too, were overcome with emotion.

Late in the afternoon, President Reagan and his wife Nancy attended the ceremony briefly, then lit a candle.

" . . . The names that are being read are of men who died for freedom just as surely as any men who ever fought for this country," Reagan told reporters in a choked voice. Asked how he thought Vietnam was going to be remembered, Reagan added, " . . . We're beginning to appreciate that they were fighting for a just cause."

The vigil, continuing by candlelight through the night, launched a week of activities here sponsored by the Vietnam Veterans Memorial Fund as a tribute to those who served in the Southeast Asian war.

Organizers of the activities, which will include a parade, dedication of the newly constructed Vietnam memorial and numerous reuinions, expect as many as 250,000 to participate.

They began gathering yesterday at the cathedral and at the memorial on the Mall to honor those who died and the thousands of others who served and lived. Some bore emotional wounds still unhealed. Many came seeking public understanding and recognition of their roles in an enormously unpopular war.

Ida Havel Bleiman came to the cathedral from Atwater, Calif. On Nov. 17, 1968, Bleiman's son, 19-year-old Richard Thomas Havel, an Army paratrooper, was fatally injured near Bongson when his helicopter landed on a mine.

"He wanted to go. He felt he had a mission there," said Bleiman, weeping. "If my boy were alive, he'd probably be going through the same thing all these other boys are going through. It makes me happy that everybody else will hear his name."

Dennis Behselich, who fought in Vietnam's central highlands with the 1st Air Cavalry, came in the uniform of an Army Reserve captain. He had traveled to Washington from his farm in Pittsville, Wis., to read the names of 276 dead servicemen.

"I hope this isn't just the vets welcoming themselves back," said Behselich. "I hope the country feels the same way."

John Philbrick, a lieutenant colonel in the Army Reserve who flew a spotter plane in Vietnam, took his lunch hour from his job at the Pentagon to be at the cathedral.

"I never had the opportunity to attend the funerals of my friends," said Philbrick tearfully. "This is it."

On the Mall, many out-of-town veterans of the war, along with friends and relatives of the war's victims, got their first glimpse of the Vietnam Veterans Memorial, itself a matter of controversy.

Several thousand people strolled in front of the V-shaped black granite structure, stopping to search for the names of sons or friends they knew who were killed in Vietnam. National Park Service guides aided them in using books that list the names of the fallen, and in locating them on the memorial's walls.

Some visitors climbed ladders to find the names. Many snapped pictures when they found the names they were looking for.

Veterans wore fatigue jackets with ribbons and unit patches from their battlefront days. The mother of one war victim, Illma Young of Oconto, Wis., wore a button with a picture of her son John: a 20-year-old with a flat-top haircut, killed Nov. 7, 1967.

She, like many others seeing the memorial for the first time, had heard of the controversy surrounding the memorial's design, the creation of a young architecture student, Maya Ying Lin. But most voiced their pleasure at the sweeping beauty of the site and the memorial.

"I'm more impressed now that I'm here," Young said. "I was quite concerned. The Vietnam war was a wrangle and now the monument. It's very pretty. It's going to stand forever."

George L. White, a retired steelworker from Lansing, Ill., his wife Selma and an aunt, Nancy Burke, together had donated $450 for the memorial and all said they were pleased.

As the sun glistened through patchy afternoon clouds, the Washington Monument and the Capitol loomed in reflection on the shiny granite tableaus. George White climbed a ladder to take a picture of his son's name.

"You don't forget," he said of Army Sgt. Gene L. White, killed when he was hit by an enemy mortar shell near the Cambodian border in 1969.

Later, a singing group from Skidmore, Mo., stood before a hushed crowd and sang "America the Beautiful."

Memories and camaraderie proved to be powerful lures, drawing many from distant places.

Charles Cowles, a retired Army medic, flew from Fairbanks, Alaska. Rick Kowalker, an unemployed former Marine machine gunner, appeared at the cathedral in blue jeans and a buckskin hat. He had hitchhiked in 18-degree weather from White River Junction, Vt., with a backpack and his 7-month-old golden retriever.

"All the guys I loaded on helicopters, I never knew what happened to 'em. I'm just paying respects to the guys who didn't make it."

Bill Crawley, a Marine intelligence sergeant in 1966, hitchhiked from Arabia, N.C., to light a candle in the chapel when the first names were read: "I knew I wasn't going to be able to get up there and read a name."

Yet the emotions of the moment were expressed most vividly in the faces and voices of those who had lost someone they loved, and still sought to find meaning in his death.

"I don't ever see us getting a chance like this again to honor my son," said Alfreda Dominique of West Palm Beach, Fla., whose 18-year-old son Gary, a Marine infantryman, was killed south of Danang 12 years ago. "A mother don't ever forget."