Audie Murphy, one of the most decorated soldiers of World War II, was awarded almost every ribbon and medal available. His name can be found on a commemorative postage stamp, a veterans’ hospital and even the Hollywood Walk of Fame.

But Mr. Murphy’s home state has never bestowed its highest military award, the Texas Legislative Medal of Honor. And for the second time in two years, an effort to give him one has fizzled.

Now family members and supporters are wondering if Mr. Murphy, who died in 1971, has been forgotten, along with other war veterans from what has been called the greatest generation.

“I’m disappointed,” said Nadine Murphy Lokey, 82, Mr. Murphy’s only surviving sibling. “I think they had him in the history books at one time, but they’ve taken him out,” she said. If students do not learn about him “and people don’t talk about him, well, they forget.”...