Yes, you can.
Coyotes are classified as Predatory wildlife in Idaho. The only firearm restriction for the hunting of Predatory, Furbearers, or Unprotected wildlife is that “no person shall …hunt with any weapon the possession of which is prohibited by state or federal law.” IDAPA 13.01.16.400.02.a. The use of an airgun to hunt coyotes (Predatory wildlife), permissible Furbearers, or Unprotected wildlife is allowed in Idaho.
The administrative rules of the Idaho Fish and Game Commission/Idaho Department of Fish and Game are found at:
http://adminrules.idaho.gov/rules/current/13/index.html The Rules for The Trapping of Predatory and Unprotected Wildlife and the Taking of Furbearing Animals are found in IDAPA 13.01.16.
IDFG identifies prohibited ‘methods of take’ for furbearers, predatory or unprotected wildlife in IDAPA 13.01.16.400 which reads as follows:
400. Methods Of Take.
01. Furbearing Animals. No person shall take beaver, muskrat, mink, marten, or otter by any method other than trapping. In Valley County and portions of Adams County in the Little Salmon River drainage, red fox may be taken only by trapping. (5-3-03)
02. Hunting. No person hunting permissible furbearing animals or predatory or unprotected wildlife shall: (7-1-93)
a. Hunt with any weapon the possession of which is prohibited by state or federal law. (7-1-93)
b. Hunt with dogs unless they comply with IDAPA 13.01.15, “Rules Governing the Use of Dogs.”
c. Hunt any furbearing animal with or by the aid of artificial light. (4-7-11)
d. Persons may hunt raccoon with the aid of an artificial light without a permit from the Director but no person hunting raccoon at night shall: (4-7-11)
i. Hunt from a motorized vehicle. (7-1-93)
ii. Use any light attached to any motor vehicle. (7-1-93)
iii. Hunt on private land without obtaining written permission from the landowner or lessee. (7-1-93)
03. Trapping. No person trapping furbearing animals or predatory or unprotected wildlife shall:
a. Use for bait or scent, any part of a domestic or wild origin game bird, big game animal, upland game animal, game fish, or protected nongame wildlife. (4-7-11)
b. Use any set within thirty (30) feet of any visible bait. (4-6-05)
c. Use a dirt hole ground set with bait unless the person ensures that the bait remains covered at all times to protect raptors and other meat-eating birds from being caught accidentally. (4-7-11)
d. Use live animals as a bait or attractant. (4-6-05)
e. Place any ground, water, or other sets on, across, or within five (5) feet of center line of any maintained public trail. (4-7-11)
f. Place any ground set on, across, or within any public highway as defined in Section 36-202, Idaho Code; except ground sets may be placed underneath bridges and within and at culverts that are part of a public highway right-of-way. (4-7-11)
g. Place any ground set incorporating snare, trap, or attached materials within three hundred (300) feet of any designated public campground, trailhead, or picnic area. (4-7-11)
h. Place or set any snare without a break-away device. (4-7-11)
Because airguns are not prohibited as a method of take for Predatory, permissible Furbearers, or Unprotected wildlife, the use of an airgun to hunt coyotes is allowed in Idaho.