Could this instrument test for vertigo in any way ?
Not really, I'm, afraid ! It's testing the balance of the muscles actually moving the eyes in their sockets.
Vertigo is a SYMPTOM (ie something the patient senses and reports, as against a SIGN which is something that an outsider such as a doctor can elicit independently).
Many lay people know there is an association between the symptom of vertigo and a sign called NYSTAGMUS, in which the eyes flick rapidly back and forth horizontally. But this is usually mediated in the Central Nervous System, not specifically in the eye muscles.