Not sure about a special draw hunt. But Utah does allow for additional help to hunters with disabilities on all hunts they offer.
Hunters with disabilities
Utah Admin. Rule R657-12
Utah provides special hunting accommodations
for people with disabilities.
These accommodations include the
opportunity to hunt with a companion, use
a crossbow, hunt from a vehicle, use a draw
lock and use a scope on archery equipment
or a muzzleloader.
Hunters with disabilities may also apply
for an extension to their season if they have
a 2011 big game hunting permit in their
possession and they meet certain criteria.
To learn more about hunting accommodations
for people with disabilities, please
visit wildlife.utah.gov/disabled/hunting.html,
wildlife.utah.gov/rules (see Utah Admin. Rule
R657-12) or call any Division office.