Since very few person will be able to have eye tracking in the future, i would suggest to implement a more natural automatic eye movement.

I think that eye movement can add a very nice impression of life in as Avatar that just look like doll sometime. (the bigger the eyes are, the more it look like a plastic face with no life)

Proposition Example:

I would suggest to use a technique similar to VR Chat.

-The eye look in the same direction even if the head move, then every 2-3 second the direction is updated by the current head direction.

-Of course if the head move to much, the eye are also updated.

-I would add an on/off checkbox (off by default) in the sdk to not break actual avatars.