Every card that supports at least Shader Model 4.0 (aka DX10 stuff) will run UE4, question is how fast enough which is subjective matter, but i think it should be OKish atleast
I have a R9 270X and am having a lot of problems even with the latest version of the engine. The whole Radeon 200 series seems to cause frequent crashes and some features like distant field soft shadows on directional lights are not working at all. As far as I read it’s caused by the drivers and it’s up to AMD to fix them, which doesn’t seem to be happening anytime soon.
Unfortunately I don’t know of any good alternative in the same price range. The GeForce GTX 770 might be worth a look.