Foxtrot... I guess the 64,000 dollar question to you is,
Do you reload ? Or better yet...
Do you care about pelt damage on a fox ? If your answer is "NO" use your deer rifle or whatever you currently own
There is not one single cal. that will do it all without pelt damage. Yes I personally have shot factory 50gr v-max's out of a 22-250 and aimed low to the eye's on a fox and hit them in the mouth with descent luck. Body Shots... Forget It... Big Holes... But again ask yourself if you kill 5-10 foxes a year do you really care ?
IMO when I see a guy ask a question like yours he can probably only afford 1 rifle for both predators which there is nothing wrong with that... But there is some trade off's with 1 rifle there will be pelt damage. So this is how I would look at it...
22-250/223/204 = Dead Coyote + Dead Fox (Pelt Damage on Fox)
22 Hornet/22 Mag/17 HMR = Dead or Wounded Coyote + Dead Fox (Minimal Pelt Damage on Fox)
I don't know about you but run offs SUPER SUCK !!! So what do you think the ethical choice would be...
I would buy a rifle geared towards coyotes and use it for foxes too and live with the pelt damage until down the road you might decide you need a rifle more dedicated for fox hunting and then you will have the best of both world's.
If you reload you can go with reduced loads and soft point bullets that work excellent on fox's... (Example= 22-250 or 223+ Blue Dot Powder + 45gr SP Hornet Bullets = Minimal Pelt Damage)
Coyotes are kinda like deer, they are pretty tough critters where fox's are built like a house cat... thin skinned and somewhat fragile...
Most store bought ammo for varmint cal. are centered around high frag bullets that will upset violently... (Pelt Damage)
Again, Reloading opens a whole new world of making one caliber more multi purpose like you want.
FYI...I have a 22-250 that I shoot Blue Dot loads though and they just shoot 5 or 6 inches low (I forget offhand) to my full house loads, so its easy to dial up 5-6 moa on the elevation and kill foxes and if I what to hunt coyotes just dial back down 5-6 moa and we are back to hunting coyotes. Never used it a lot because I use a Hornet for night hunting foxes and the Hornet is easier to carry too. Still very effective, quiet and very deadly load.
17 caliber stuff that I have seen is still hard on fox, except the 17 HMR which I've seen many bullets splash on the face/head when night hunting and that makes for run offs or the game of can you catch the wounded fox when they run after you walk up to them (Very Very Fun To Play)
...
Just think your choices through and buy the best gun you can afford and not the cheapest gun you can find in that caliber... Buy good stuff and get good results, you will be much more happy in the end and not selling guns,scope bases,scopes and being very frustrated.
Good Luck