It appears that he did actually make the trip, starting in southern California on September 8, 1982, and ending in Washington, D.C., on May 14, 1986.
The Los Angeles Times ran a short blurb back in 1986:
Wieland, a former combat medic who lost both legs in a 1969 land mine explosion in Vietnam, began the trip on Sept. 8, 1982, at Knott's Berry Farm near Los Angeles. His parents and friends walked the final mile with the exuberant, muscular veteran, cheering him on to the memorial on the Mall.
A UPI wire story from that time also covered his accomplishment, in detail (q.v.):
A legless veteran who hobbled 2,000 miles across the United States on his padded knuckles ended his four-year trek Wednesday with an emotional ceremony at the Vietnam Veteran's Memorial.
A contemporary story from the Chicago Tribune explains some of the logistics of the trip (but little else about it):
He was blessed with an occasional helper, but most of the trip was a solo affair. That entailed "walking" a prescribed distance to his van, then driving back to the point where he'd left his wheelchair, then advancing a few more miles by motor, then parking his van, then taking his wheelchair back to repeat the entire procedure. It took him almost four years.
A 2011 retrospective from the New York Times gives little detail about the journey, but affirms that it took place.
Most sources also mention that on the day his walk ended, he visited President Reagan at the White House.
A video is available on YouTube showing some contemporary footage and demonstrating how he walks with his hands.
Finally, I was able to find an article written by someone who had met him on the journey. It was written more than a decade afterward, and a few of its details are not consistent with contemporary reports.