The man, identified as Harry Whittington, 78, is reported to be in good condition in a Corpus Christi hospital, after Mr Cheney sprayed him with shotgun pellets at the Armstrong Ranch, according to property owner Katharine Armstrong.
She said the vice president turned to shoot a bird but accidentally hit Mr Whittington instead.
The lawyer from Austin, Texas, was mostly injured on his right side, with the pellets hitting his cheek, neck and chest.
Mr Cheney's spokeswoman, Lea Anne McBride, said he has spent time by Mr Whittington's bedside.
The local sheriff's department is investigating the incident.
Cheney "apologetic"
Ms Armstrong told AP that she was watching from a car as Mr Cheney, Mr Whittington and another hunter got out of a vehicle to shoot at a convoy of quail.
She recounted that Mr Whittington shot a bird, and later approached the other two hunters "and didn't signal to them or indicate to them or announce himself".
"Fortunately the vice president has got a lot of medical people around him and so they were right there and probably more cautious than we would have been," she said, adding that Mr Cheney was very apologetic.
"This is something that happens from time to time. You know, I've been peppered pretty well myself," said Ms Armstrong.