Bob Dole went from the plains of Kansas to the battlefields of Italy, where he was left for dead with grievous wounds, before a dogged recovery enabled him to become a widely respected leader of the Senate and Republican nominee for both president and vice president.

He died early Sunday morning in his sleep, the Elizabeth Dole Foundation said. He was 98 years old. House Speaker Nancy Pelosi ordered the flags at the U.S. Capitol to be flown at half-staff in remembrance of Mr. Dole.

President Biden on Sunday praised his long friendship with Mr. Dole, saying they were able to work together in the Senate even when they disagreed. “Bob was an American statesman like few in our history,” Mr. Biden said in a statement. “A war hero and among the greatest of the Greatest Generation. And to me, he was also a friend whom I could look to for trusted guidance, or a humorous line at just the right moment to settle frayed nerves.”

Mr. Dole was a fixture on the Washington scene for more than half a century and a national leader of the Republican Party for nearly as long. As a legislator, and ultimately as leader of the Senate, he played a role on a staggering list of legislation touching every aspect of American society: voting rights, Social Security, food stamps, child-nutrition programs, the rights of the disabled, the North American Free Trade Agreement and more. As Congress’s chief tax writer, he was instrumental in the landmark Reagan-era tax cuts as well as in an overhaul of the nation’s tax code in 1986.

Over the arc of a 36-year career in Congress, Mr. Dole underwent a steady but dramatic transformation. He once was seen as a partisan slasher, a reputation enhanced by his time as Republican National Committee chairman when he was a fierce defender of former President Richard Nixon during the Watergate crisis. In time, though, his partisan edges softened, and he worked with liberal icons George McGovern and Ted Kennedy on major legislation.

For two decades he also was in the thick of Republican contests for the White House, starting with his nomination as Gerald Ford’s running mate in 1976 and proceeding through unsuccessful quests for the Republican presidential nomination in 1980 and 1988. He finally got the presidential nomination he had long craved in 1996, but it was a poor time to carry the GOP banner. Democratic President Bill Clinton was seeking re-election amid a period of peace and prosperity, and the incumbent won handily.

Though Mr. Dole had promised that he would return to his Russell, Kan., roots if he lost the race, that didn’t happen. He was far too much a fixture in the nation’s capital by then, both in his own right and as the spouse of former senator and Transportation Secretary Elizabeth Dole. So he became a high-powered member of Washington law firms, working for clients such as the government of Taiwan.

In his later years, he became best known for championing causes near and dear to him and other members of the World War II generation. He was a driving force behind creation of the World War II Memorial on the National Mall and then adopted the cause of Freedom Honor Flights, organized trips to Washington for World War II veterans so they could visit the memorial built in their honor.

There was a bond between Mr. Dole and his fellow veterans, often an unspoken one. “I’ve been knowing him for a long time,” said one veteran who met Mr. Dole on a Freedom Honor Flight in 2009. “Of course, I’ve never met him before.” In many respects, Mr. Dole became the personification of the Greatest Generation, the one that fought and prevailed in World War II.

Mr. Dole was born in the small western Kansas town of Russell on July 22, 1923. His father, Doran, was a man of few words who ran an egg and cream stand, clad always in overalls. (Mr. Dole would for years keep a picture of his father in those overalls in his office.)

The family scratched out a living in Russell. Mr. Dole’s mother sometimes sold sewing machines door to door, and young Mr. Dole washed cars, delivered newspapers and worked as a soda jerk at the local drugstore.

Mr. Dole also was a good athlete who lettered in three sports in high school, a record that helped lead to a big leap on his part. He enrolled at the University of Kansas and sought a spot on the powerhouse Jayhawk basketball team under legendary Coach Phog Allen. He earned a spot on the freshman basketball team in 1941 and made the cut for the varsity team a year later.

But war soon superseded basketball. He joined the Army, had hopes of becoming a medical doctor and trained in medical school for a time. But the Army instead sent him to Officer Candidate School in Georgia and made him into a second lieutenant in the infantry, specifically in the 10th Mountain Division. The young man from the flat plains of Kansas was an odd fit in a division meant to be made of skiing soldiers.

Late in the war, in 1945, Mr. Dole’s unit took part in a spring offensive in Italy, during which his platoon was assigned to take a hill that lay across a mine-laden field covered by enemy snipers. In an exchange of fire, he was hit in his right shoulder by exploding shrapnel. Two rounds of medics were gunned down trying to rescue him before he was pulled back to safety.

His fellow soldiers assumed he wouldn’t survive his wounds—and he nearly didn’t. Eventually he was sent back to the U.S. in a full body cast and spent 39 months in recovery. He had lost all mobility in his right arm and hand, and simply getting dressed in the morning would be a challenge for the rest of his life.

At an Army medical center in Michigan, in his final months of treatment, Mr. Dole met a young occupational therapist named Phyllis Holden at a hospital dance. She became his wife a few months later, and the newlyweds returned to Kansas, where Mr. Dole earned a law degree at Washburn University.

He made his turn to politics by winning election as county attorney and then ultimately to the U.S. House of Representatives in 1960. He worked on farm issues and rural electrification and never missed a vote in the House.

His chance to move up came when a U.S. Senate seat opened in 1968. He ran and won. He was re-elected in 1974, in an especially nasty election against Rep. Bill Roy in which Mr. Dole started out well behind. He went on the attack, most notably calling Dr. Roy someone who delivered “abortions on demand.” He won narrowly, and his seat was never in jeopardy again.

Mr. Dole divorced Phyllis in 1972 and married Elizabeth in 1975.

Throughout his career, Mr. Dole was unfailingly loyal to the Republican Party that had nurtured him, a loyalty that led him to defend Mr. Nixon without hesitation and, years later, to support Donald Trump as Republican presidential nominee in 2016 at a time when many other party veterans were refusing to do so.

“A bright light of patriotic good cheer burned all the way from Bob’s teenage combat heroics through his whole career in Washington through the years since. It still shone brightly, undimmed, to his last days,” Senate Minority Leader Mitch McConnell of Kentucky said in a statement.

Over time, he healed some lingering political wounds. President Bill Clinton, who had defeated him handily in 1996, gave him the Presidential Medal of Freedom the following year. In addition, he and former President George H.W. Bush, a fellow World War II veteran, buried the memories of the nasty primary campaign they waged against each other in 1988 to become friends who joshed with each other about their ages and frailties. Later, Mr. Dole rose from his wheelchair to deliver a memorable salute to the late Mr. Bush as he lay in state in the Capitol.

—Stephanie Armour contributed to this article.