What distance are you shooting at? that could influence a particular caliber selection.

I don't live in PA though....

I'll zero at either 100 or 50 yards and probably be shooting from about 30 yards (if I put my e-caller 30 yards away and they come STRAIGHT into it) out to around maybe 125.

The 22 Mag would be good for your smaller animals within your ranges and okay for the yotes if you're able to call them in close like that. If you think you're going to be shooting at the longer ranges, NoMercy's suggestion about the 22 Hornet would be a better fit.

Since you haven't mentioned reloading, I assume you're not reloading. 22 Magnum ammo is cheaper than 22 Hornet ammo but it's not as versatile, especially if you decide to reload. Something else to consider is how many rounds do you plan on firing in an average year? Yes, 22 Hornet costs more but you get incredible versatility in a relatively quiet round that will easily cover all of your anticipated ranges. If you only anticipate shooting a few boxes of ammo a year, the cost difference isn't that great for what is arguably a better round for all of your chosen animals.

A perfect solution would be to buy both but since your funding doesn't allow it at this point, I'd lean towards the 22 Hornet and pick up the 22 magnum later on when your budget allows it.

I'm not really worried about ammo costs but I'm worried the .22 Hornet will blow the smaller things (groundhogs, beavers, coons) into pieces or destroy their pelts beyond repair.
I can probably call yotes in to about 60-70 yards. We don't have many here anyway, so since they're not main prey, I'd rather not be able to shoot coyotes than ruin pelts on coons/hogs/beavers.
Will the .22 Hornet destroy smaller animals?
With this in mind, should I get .22mag or .22 Hornet? I think this'll be the final decision.

It looks like you're narrowing things down to groundhogs, beavers and coons and they will all likely be at shorter range (70 yards or closer). Since preventing pelt damage on those smaller animals seems to be your primary issue, you're not reloading and ammo costs are not a factor, it looks like 22 Mag would do just fine. If those pelts are profitable enough, you could earn enough to pick up a 22 Hornet later and be covered for everything.

There are other guys on this board who have extensive experience with all of the animals you're mentioned and could fine tune your selection even better, especially with ammo selection. My experience is more with coyotes on up and at longer distances out here in the West. I rarely see foxes and bobcats and they require something smaller in caliber than a yote. Not to throw a wrench into your caliber selection plans but my next purchases will likely be a 17 Hornet and 17 WSM for shooting bobcats and foxes with enough steam left for coyotes when needed.

Well, I'm not shooting many yotes, but I still want enough gun to shoot a bobcat or fox. Will a .22Mag kill those? If not, I'll go Hornet and deal with the pelt holes and sew them up on the smaller animals.
I'm leaning towards .22 Hornet. Is .17 Hornet an option for me?

The 22 Hornet is on the ragged edge bottom for coyotes. It's about ideal for cats, fox, coons, beaver, badger, and similarly sized creatures. 22 Mag is even further down on the ragged and tattered edge for coyotes, and will be even nicer to the other game.

There's a painful reality with which you'll have to come to grips, however - groundhogs are d@mned small animals. I'm not sure why in the world anyone would put in the energy to tan one. Even a 22mag can make a mess of such small rodents, so the only real option there, if you want pristine hides, is 22LR or 17HM2, which will put you firmly too small for coyotes, and only suited for anything else on your list at the shortest of ranges. You're talking animals which span from 5-6lbs average up to 40lb+ animals, and based on how these are hunted, you could be talking about 20yrd shooting out to 200yrds or more... Anything which doesn't destroy a groundhog is NOT going to be effective at killing a coyote, and marginal for the rest of your list. Anything highly efficient at killing badger, coon, & beaver sized game will be too much for the groundhogs and too light for the coyotes. Anything really effective at killing coyotes will be over powered for the rest of your list, and will blow groundhogs to bits with anything except for ball ammo (not legal for use in all states).

So personally, I'd pick up a 22LR for the groundhogs and a 22Hornet for the rest, and you'll be quite well suited.

17 Hornet is an option, but is less effective on coyotes, and the higher velocity, lighter constructed 17cal bullets are prone to do worse damage on smaller game like groundhogs.

The .17 Hornet might be Your best bet for taking down the many Furbearers Your targeting...as several mentioned Your choosing a pretty broad selection of Animals that might require a little larger Caliber especially when getting up to Foxes & Coyotes.

I have a .22 Hornet and have shot several Groundhogs and it didn't seem to do a lot of pelt damage...but I didn't skin it out so I couldn't say for sure how bad it was?I can say with the Hornady 35 grain ballistic tips it only had a small entrance hole and NO exit hole!

I'm still waiting to call in a Coyote and put it down with my Browning Micro Hunter .22 Hornet...personally I think it will do the job with good shot placement and under 125 yards,if not I rely on my .204 or .223 for larger Predators.