rather than answer the question for you since I am not there to look your place over I think I will give you the info you need to know to make the decision
your options being shotgun or rifle
shotgun you have some ammo choices shot is lots of small projectiles the larger shot is called buck shot buck shots most common form is 00 called double ought and is typically comprised of nine 30 caliber balls loaded in a shot shell these balls spread out based on the choke / restriction at the end of the barrel full being the greatest restriction commonly approved for use with normal factory chokes , there are specalty chokes that allow for more
when we shoot multiple projectiles IE shot we talk about pattern this is most often discussed in terms of a 30 inch circle and the number of shot that fall inside the 30 inch circle when you shoot at the center the other thing to consider is the size of your target will the spread of your shot be wide ebought at your target could slip between the pattern.
if you look at my post about suprema buck shot I tested with a modified choke I didn't have a full at the time I expect that I could extend my effective range from 30 yards to 40-45 yards with a full choke
do not confuse effective working range with distance that it could go and do great bodily harm the load runs out of effective usefulness long before it stops being dangerous
round balls have the worst ballistic coefficient of all projectile shapes , this is they push air with as much surface area as is possible for their weight an easy example of this is give from round cannon balls vs cylindrical cannon projectiles , looking back at history Fort Polaski Tybee Island Gorgia was shelled round cannon balls traveled the mile and a half and bounced off the thick brick walls , canonical "bullet shaped" cannon fire however could extend range to 4-5 miles and the energy carried when it got there was 4 to 5 times greater and would cause much damage to the brick walls of the fort and the fort was surrendered by the confederacy rather than loose it and all the men there were many forts built in the day or round ball cannon fire that were crushed by the new cylindrical projectiles at never before heard of ranges
so now we know why a round ball might be desired if the range your willing for it to be dangerous at is shorter
so how does one calculate round ball ballistics
weight in grains \ (10640 x diamiter in inches )= balistic coeficent
to simplify a .315 round ball weighs 47gr or 47/7000 of a pound as there are 7000gr in a pound
47gr \ (10640 x .315)= .014
you can then plug the balistic coeficient into the calculator
http://www.hornady.com/ballistics-resource/ballistics-calculator
the weight of 47gr and the sight in range of 25 yards set the calculator to extrapolate every 25 yards and you can see that given an average muzzle velocity of 1200fps at 125 yards the round ball even if fired from 50 inches high and level should be in the dirt you want to probably double that distance to anything you might hit for safety and even more if that is a person
shooting down and at targets no farther than 35 yards will put your round int he dirt sooner , smaller shot will also put your round in the dirt sooner , but you get less energy to the target
rifles go a lot further competition distances for high power rifles are 1000 yards and the round can fo even further , slugs also go further than shot ussualy they would be in the dirt in 400 yards even if fired level but they run out of effective accuracy around 150 yards depending on the slug ,modern saboted slugs with much greater ballistic coefficient really just need to be treated as rifles
a shotgun is a very effective tool for close targets inside 40 yards , inside 100 or so with slugs but keep in mind the range they can go
another option may be to trap them , some will use a 22lr rifle as they have less range , the way they are use is important anything shot down into the dirt will be safe at much shorter range behind the target
so ask yourself how close have i seen the coyotes , have you seen the coyotes ?
traps may be the better answer if you can get them in a trap you can dispatch with a 22 or a shotgun form very close range at a downward angle and while you wouldn't want to stand behind the yote when someone is dispatching at a distance of a few feet I have done it with buildings as little as 10 feet behind and never had a problem but i am shooting down it is dirt and not a hard flat surface like concrete between me and the building and I get close like almost touching them with the barrel keeping a good down angle
should you have a gun , if you feel it is time , then yes
should it be a shotgun , probably
should you shoot buckshot where you live , only you can answer that
smaller shot goes less far but it retains less energy for the power to cleanly dispatch also
my brother lives on 3 acre , there are places where i could feel fine taking a shot and a bunch where a shot should not be taken
so are you going out to find coyotes still in the coop or do you just see them running at the property line or do you not see them at all
a trap can work 24/7 and you probably can't wait and watch 24/7 as there are likely other tasks you must do in a day