I have problems as you can chick in the uploaded video with the pupil's movement. only the right pupil moves left and right while the other doesn't move at all. Also, both don't move up and down. I've just started using the program and was following Okay Samurai tutorial on youtube and I'm stuck here.
Can you share a screen shot of the rigging panel with the eyes expanded? it is usually a tagging issue. But it is hard to tell without seeing the artwork and your puppet hierarchy.
I wrote up this blog for the most common eye problems I have seen. There are other ways to do eyes, this is just the pattern I always follow (because it generally works for me!): https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/