Awakened plants may or may not have eyes, but a medusa's Petrifying Gaze only requires that
a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa
As you say, the Awaken spell grants its target
senses similar to a human's.
This would certainly suggest that it would be able to see. From the Monster Manual (page 8), we have this:
A monster with blindsight can perceive its surroundings without relying on sight, within a specific radius.
Creatures without eyes, such as grimlocks and gray oozes, typically have this special sense, as do creatures with echolocation or heightened senses, such as bats and true dragons.
If a monster is naturally blind, it has a parenthetical note to this effect, indicating that the radius of its blindsight defines the maximum range of its perception.
Most plant-based creatures (for example, blights, or everything in the Fungi section) follow this exact pattern, having blindsight X ft. (blind beyond this radius). The Awakened Tree and the Awakened Shrub have no blindsight, so either they are actually incapable of perceiving their surroundings at all, or they are simply using ordinary sight and are therefore valid targets for a medusa's Petrifying Gaze.