Ground Controller reported he failed to properly coordinate a snowplow with the Tower Controller; resulting in a runway incursion.

Narrative

Prior to the event; the field had been IFR for most of the day and the snowplows were out brushing the runway. During the time of the event; it was VFR. Unit XA called me up and told me to show him as part of Unit XC. I queried him on his position on the airfield and he told me at the North Deice Pad. What I failed to realize was that Unit XC had been approved on Runway 2L with Tower. I told him 'Roger; Unit XA; I will show you as part of Unit XC and company.' A few minutes later; Tower was telling Unit XC to exit the runway for a departure. She saw there was another snow plow on the runway and said Unit XC and company plus 1 exit Runway 2L. This is when I realized that I should have told Unit XA to hold short of Runway 2L and contact Tower. Part of the event was my fault; but on the other side of it; I never verbally approved Unit XA on Runway 2L. All in all; the event was very minor; but it alerted me and in the near future; I will be careful when approving vehicles to be apart of other companies.The biggest lesson I learned is scan scan scan and understand what a vehicle is asking before approving or saying roger. This would have prevented the vehicle from going onto the runway without approval from Tower.