What does the law allow in terms of firearms? As far as the how, well it ain't rocket science with coyotes. Either call them in, or sit patiently on your deck, have a snack, and wait for one to appear. Looks like you've got some pretty good sight-lines based on your avatar pic.
Hunting Hours and Methods
•Bobcat, coyote, red fox, grey fox, raccoon, opossum, and skunk may be hunted at any hour, day or night, beginning after sunrise on the opening day of the season and thereafter until midnight on the closing day of the season. For more on hunting furbearers at night, see below.
•You may hunt furbearers using a handgun, shotgun, rifle, bow, or crossbow. All laws pertaining to the use of firearms apply.
•Crossbows may not be used while hunting with a dog for any small game, except for coyotes, in the Northern Zone.
•You may hunt furbearers with a rifle chambered in any cartridge, except that during any open season for deer - including archery, muzzleloading and regular seasons - you may not possess a rifle larger than .22 caliber rimfire afield, during the day or night, to hunt wildlife, including furbearers in any county or portion of a county where deer hunting with rifles is prohibited during the regular deer season. Centerfire rifles smaller than .22 caliber are permitted.
•Air guns may be used to hunt furbearers. An air gun is defined as a firearm that uses spring or compressed air (not gunpowder) to propel a single projectile that is .17 caliber or larger and produces a muzzle velocity of at least 600 feet/second.
•You may use a call, including an electronic call.
Furbearer Hunting at Night
You may hunt furbearers at night, with or without a light, as follows:
•You may use a light, but you may not hunt from any motor vehicle, including an ATV. All laws pertaining to the use of a spotlight and firearm apply.
•If hunting without a light, the use of a light gathering ("starlight") scope is legal on any firearm listed below.
•You may use a handgun, bow, or crossbow.
•You may use a shotgun loaded with shot (any size).
•You may hunt furbearers with a rifle chambered in any cartridge, except that during any open season for deer - including archery, muzzleloading and regular seasons - you may not possess a centerfire rifle afield, during the day or night, to hunt wildlife, including furbearers in any county or portion of a county where deer hunting with rifles is prohibited during the regular deer season.