What kind of action? So far, much of what's been discussed is bolt action fodder that's been on the shelf for decades. A lot of it is quite good for deer, but none is different or very advanced.
Now,a hunting AR15 with fixed stock, tube handguard, and a red dot, in 6.8/6.5, would be different. If you reload, ammo wouldn't be hard for the 6.5, more is on the way. If getting it now before the season, 6.8 could have you shooting in a few days. A good case can be made for either, they share low recoil, low carry weight, a wide choice of optics, are extremely durable, have twice the foot pounds of force of the 5.56, are legal hunting calibers almost everywhere, and allow a great deal of customization with features and color. For hunting, the AR is does very well putting a bullet on target.
Or, a medium recoil bolt gun in a Walmart caliber like everyone else recommended, which seems to be a bit contrary to what you asked. But, to be quite honest about it, the deer would be as dead. It's really up to you.