A disabled Scout who wishes to earn his Eagle must petition the Council (or District?) Advancement Committee on the basis that his disability prevents him from completing the requirements. The Committee consults with the Scout, his parents, and any medical personnel and records as appropriate. The Scout and his leaders and the Committee then work out a plan that accomplishes the same ends, but represents something the Scout can actually do. IIRC, the Scout must complete those requirements of the usual required merit badges that he can. It is also possible, especially in the case of mentally handicapped Scouts, to waive the age requirement (normally an Eagle Scout must complete all requirements for the Eagle badge prior to his 18th birthday).
"It took him longer than some of the other boys, but he earned every badge the hard way," said Roe's father, Arne Roe, an assistant scoutmaster for Everett Boy Scout Troop 114. "When he earned his wilderness-survival badge, he had to build his own shelter out of sticks and branches and he stayed there all night. He really has got perseverance."