Autofocus systems need a certain minimal amount of light to operate. It doesn't matter what the source of that light is as long as the light is present at the time the camera attempts to autofocus.
The light you've linked is a continuous light source. That is, it provides light continuously before, during, and after the photo is taken. Even in its self-described "flash mode", it's operating as a continuous light for approximately one-half second.
If you need AF assist using light from the ring you will need to switch the ring light to the mode that leaves the light on continuously so the light will be on before you fully depress the shutter button.
You could also use any other light source that is shining on your subject to illuminate it. Flashlights can come in handy for this. Many who do very long exposure photography in the dark with subjects too far from the camera for the camera's AF Assist light to reach it with sufficient intensity (if the camera even has one) will use a flashlight to AF or manually focus, then hold focus and turn off the flashlight before beginning the exposure.
As long as the intensity of the continuous ring light is sufficient at the distance you're using it the camera's AF system will be able to use it to focus. The ring light should be much brighter than the camera's AF Assist lamp, even at the lowest intensity setting. Of course that also assumes your target has sufficient contrast running in a direction that the camera's AF points can use, the same as would be the case even in daylight conditions.
If you try to use the "flash mode" the light won't come on until you fully press the shutter button and the exposure has already begun. Since it's not a TTL flash it will not be able to sense when the shutter button is half pressed and turn the light on at that point.
- You have the camera set to not release the shutter until AF is confirmed
- The camera can confirm focus and take the photo in the one-half second window the light is turned on
it won't provide AF assist since the light depends on the flash sync signal from the camera's hot shoe to tell it to turn on. The camera won't send the flash sync signal until focus has been confirmed and the first curtain of the shutter is fully open. Even shooting in Live View the camera won't send the flash sync signal until after focus is confirmed.
If you have the camera set to release the shutter without confirming AF by the time the light comes on in response to the sync signal the image is already being exposed and the AF system is not active any more.