How do I watch Netflix on a non-smart Vizio TV?

Here are several answers--but keep in mind, they all involve attaching a device of some sort--whether it's a computer, a laptop, a tablet, a phone, or a device like Google's Chromecast, Amazon's Fire Stick or FireTV, or other devices similar to those I just mentioned from Google and Amazon.
You will usually need an HDMI cable to attach your device to the TV (whether it needs an HDMI cable or not depends on whether the device plugs directly into your TV's port or whether it needs a separate cable--of which there are standard HDMI, mini HDMI, and micro HDMI).
The simplest, easiest, and least expensive types of equipment to enable your non-"Smart TV" to view Netflix are the "key-fob" type devices like the Chromecast and Amazon Fire Stick (there are others, but these are the most common, most popular, and well-supported devices.)
You would simply plug the device into your TV's HDMI port, attach the power cable (these devices may be able to draw power from your TV's HDMI port, but often require a separate power adapter to be plugged into them) log onto your Google Play account or Amazon account, download and install Netflix, and log onto Netflix. Usually (but not always) the Chromecast and Fire Stick have Netflix installed, but you may need to update or download and install/reinstall the Netflix app.