Definitely not my area of expertise, but coax (to the best of my knowledge) is analog only. I don't believe you can convert it to digital.
Your cable box takes the analog signal, interprets what you have converts to a digital output.
I'm making an assumption here, but your believe is if you can connect the coax to the TV via HDMI you can view/control the channels you receive? It doesn't work like that. The cable box is a key component as far as what channels you can receive/view.
Is there a substitute for the cable box that would do the same thing? Currently the cable goes direct into my Vizio TV that has the coaxial input and delivers the channels. With this new TV I am looking to find someway to connect the cable without having to rent another cable box. Will a DVD player or some other device substitute?
My new TV is a VIZIO SmartCast 50" Class E-Series - 4K Ultra HD, Smart, LED Home Theater Display - 2160p, 60Hz (E50x-E1). I don't want to have to rent another cable box. There has to be some way to connect my cable input to the HDMI input on this TV?
If you are trying to watch digital cable tv you need a cable box. There are no legal ways around this. They do more than convert coax to hdmi, they are a descrambler. In the old days with basic cable you could get basic cable by plugging it directly to the tv with no cable box but those days have long since passed and digital cable requires a cable box.
As soon as you said RENT CABLE BOX. This signifies a cable service provider is involved. They ENCRYPT the signal sent to you and ONLY THEIR BOXES can decrypt that signal (they are no dummy). On the same cable they can send you QAM (typically local broadcast channels), and this is not encrypted but at least in the U.S., basically QAM is gone, you don't say where you are posting from.