With a lifetime of musical activism and nonviolent resistance behind her, Mavis Staples still sings for peace and justice.

August 14 at the Martha’s Vineyard Performing Arts Center, Ms. Staples—now 79—delivered a fiercely inspiring performance of protest and inspirational songs, spanning a career she began by singing with her father, brother and sisters at Chicago-area churches in the 1940s.

Taking the name Staple Singers in the 1950s, the Staples family entered civil rights history when they became the musical opener for Dr. Martin Luther King Jr. in the 1960s, singing before he spoke at rallies and marches.

The Staple Singers also made their mark on popular music with message-driven hits like I’ll Take You There and Respect Yourself, both released in 1972, and (If You’re Ready) Come Go With Me, from the following year.

The family’s well-chosen covers included the Band’s 1968 classic The Weight, Buffalo Springfield’s 1967 rock protest anthem For What It’s Worth and Slippery People, the Talking Heads song from 1983. And Ms. Staples sang them all at the PAC, along with new songs from her latest album, If All I Was Was Black (2017) and a cover of the 1971 Funkadelic single Can You Get to That.

The band she brought to the Vineyard was composed of longtime collaborators Rick Holmstrom on Fender Telecaster guitar, Jeff Turmes on bass and Stephen Hodges on drums. Ms. Staples’s supporting vocalists were Canadian soul singer Donny Gerrard and Vicki Randle, who is also a multi-instrumentalist known for being the first and only female member of the Tonight Show band.

It was Ms. Staples’ first concert on Martha’s Vineyard, for which she teasingly chided the audience.

“When I come to a place for the very first time, I wonder: What took you so long?” she said, adding puckishly “Don’t y’all try that again.”

One of the most recognizable voices in pop music for nearly half a century, Ms. Staples’ soulful growl remains undiminished, as does her zeal for both social change and human connections.

“I’m tired of us living so lonely/I think I know what to do/Gonna build a bridge over the mountain/Gonna walk right over to you,” she sang in the song Build a Bridge, from the new album.

“Can’t you feel it in your bones? Change is coming. On every walk of life, people are seeing the light,” Ms. Staples said, after bringing the audience in to sing the chorus on For What It’s Worth: “Stop, children, what’s that sound, everybody look what’s going down.”

Introducing the Staple Singers’ Marching on Freedom’s Highway, she recalled that “this is a song that we would wake up with every morning and go to bed with every night. We were singing this song all day long as we marched down the Southern highways with Martin Luther King,” Ms. Staples told her cheering audience.

“They put us in jail, but we got back out and we started right over again.”

Her father, Roebuck (Pops) Staples, wrote the song in 1962, Ms. Staples said. “He wrote it for the big march, the march from Selma to Montgomery, Alabama,” she said, and then sang gospel-style: “I was there, and I’m still here. I’m a living witness.”

Some audience members rushed to the front during the Staple Singers’ 1974 Touch a Hand, Make a Friend, when Ms. Staples reached out from the stage to clasp hands with eager listeners.

“Thank you, baby, I love you,” she said, adding “When you see your neighbor, put a smile on your face. It’s nice to be nice. Be nice, y’all.”

The concert, a 35th-anniversary benefit for WMVY 88.7 FM, began with a performance by the folk-soul trio Dwight & Nicole, who have been opening for Ms. Staples on her summer tour. Supporting their new album Electric Lights, guitarist Dwight Ritcher and bassist Nicole Nelson sang more than a dozen originals and an English-language arrangement of Edith Piaf’s 1940s song La Vie en Rose. Their drummer, Ezra Oklan, added power and depth to the duo’s sound.

Next up for the Martha’s Vineyard Concert Series is Audra McDonald on Saturday, August 18. Visit mvconcertseries.com.