Looking around with your eyes without moving your head

Is there a word or phrase for looking just by moving your eyeballs?

Something like: "the way the helmet was attached to the armour prevented her from turning her head to look at him. Instead, she looked around with her eyes and caught a glimpse of him out of the corner."

Solution 1:

They swivelled their eyes towards X imparts the meaning you want, that the movement of the eyes happens independent of the head.

Swivel-eyed refers to someone who is constantly swivelling their eyes, implying frenzy. However they swivelled their eyes will be understood as a one-off occurence.

If they have not found what they're looking for, you may use sweep or swept as in:

Their gaze swept the horizon for pirate ships

Swept implies a swift, broad and lofty movement which is much more suited to the movement of the eyes than the rest of the body.