Coyotes out to 50 yards.

Cost? to 3-5 grand, do not want to go cheap and wish Id done it right from the beginning. Don’t want to run to refill tanks safely. Rifle, tank, compressor and accessories. One and done, first time doing it right. Recommendations also for safe, quality support equipment. No one in my area sells this gear that I know of. Recommend Online dealers who can answer questions reliably when necessary.
 
Cost? to 3-5 grand, do not want to go cheap and wish Id done it right from the beginning. Don’t want to run to refill tanks safely. Rifle, tank, compressor and accessories. One and done, first time doing it right. Recommendations also for safe, quality support equipment. No one in my area sells this gear that I know of. Recommend Online dealers who can answer questions reliably when necessary.

There are companies that will give you great information on what you are looking for. I like to buy some things from pyramid air but some of the guys in their technical department aren’t as knowledgeable as they think they are.For me, I’ll buy from Baker air guns and talk to Donnie to get the advice and information that I need. I hear Airguns of Arizona also have some good people working there. There’s people here on this forum that have more experience dealing with some of these companies and can point you in the right direction Because they have dealt with so many different companies.
 
As you can now see, there are MANY airguns that will get the job done. 3-5g will get you a very nice an reliable pest controller. The rest is up to you. Everyone here has their own preference. It's now what perks you want. Some will give you one thing but lack another. Here is my setup for pigs. And believe me, it doesn't miss and get the job done easily. .30cal averaging 910 fps with 50gr pellets. Find 'em Shoot 'em Eat 'em The southern boys way to get the job done!
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I do head shots but I really do want to know what it will do on a body shot. I have a lot of pork ribs in my freezer. I will thaw and shoot a rack tomorrow to see what it will do. Pig ribs are much thicker than coyote so lets see what it will do at say 35yds? Cover them in an old t-shirt to account for skin and fur. Cantaloupe for insides maybe? Got me curious, now I want to know.
 
I am all about heads shots, I've been taken big pigs with small bores for well over a decade now,

but if I was to take a coyote I would want a big caliber to have some leeway on a moving shot,

they don't exactly stand still much and at 50 yards I would not bank on consistent head shots so yeah if that was my task a .45 or .50 Texan would come my way,...and then you can use it for deer too :)
 
I’d recommend a .30 caliber using pellets at 50 yds if you are in an urban environment where there are people living nearby. Pellets slow themselves down the farther they go whereas slugs just keep right on going and a slug’s ricochet off a rock hidden in the dirt will zing off into the sunset...not exactly what you want in a neighborhood of 1 acre lots. 
The Evol .30 is a mean machine and will take care of any coyote problem you’ve got, as will a Uragan .30. I have the Uragan King in .30 and you get a ridiculous number of shots per fill of air, and it is quiet and astoundingly accurate. It’s heavy and meant for bench shooting, so shooting from a porch would be perfect. The Evol is easy to carry and makes the same power as the Uragan, but you get about 30-35 shots per fill as opposed to the King which you fill with air and shoot until your trigger finger falls off. Both of those air guns would be “doing it right” and don’t have much room for improvement. There’s no need to start trying to tune or modify either one out of the box. Just fill them with air, mount and zero a good scope with some JSB 44gr or FX 45gr pellets and that’s it. You are ready to decimate coyotes.

As far as compressors go I have the Air Venturi 4500 compressor and it’s done well for me for a couple years now. Or... you could just rent a 6000psi nitrogen tank from your local welding supply shop and buy an adapter to fill your airguns with. It’s fairly cost effective to do that. I’m not sure of the prices now but it seems like it used to be like 50 bucks a year or so if I remember correctly. If you go that route you don’t have to worry about maintenance on a compressor and you have an ample supply of air. 
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I do head shots but I really do want to know what it will do on a body shot. I have a lot of pork ribs in my freezer. I will thaw and shoot a rack tomorrow to see what it will do. Pig ribs are much thicker than coyote so lets see what it will do at say 35yds? Cover them in an old t-shirt to account for skin and fur. Cantaloupe for insides maybe? Got me curious, now I want to know.

Well, if you aren't gonna get any ribs out, then I guess I'm gonna have to do it!
 
In a neighborhood environment, I'm not sure you'd want to do chest shots at all. Some will run over into a neighbor's yard to die and you'll have to do The Walk of Shame.

Also remember that the bigger the caliber, the louder they are and harder to quiet down. A .50 cal. will be louder than a .22 rimfire. Same for .357. A .30 cal. even will require a hefty moderator to be neighborhood-friendly.

You don't need anything more than a nice .25. Pellets will be cheap. (compared to anything bigger bore)

The only thing about yotes is that they have that angled skull, so if they're looking at you, you pretty much need to hit them right between the eyes. It's better if they're looking away, so you can put one in the back of the head or in the ear.

.25 with pellets will also do the job with a lung shot, but they will run a bit before expiring.

Something like a Benjamin Marauder in .25 with an additional moderator on it will do the trick for under a grand. Scope-wise, you don't need anything too fancy: a UTG 3-9x50 with lit reticle will do nicely. You'll want that 50 mm objective for shooting in low light, and the lit reticle for being able to see it over the yote.