By RAW, no it could not, because spirits don't have the Fly power. Air Spirits have the Flight skill which is to flying what the Run skill is to running, but the actual ability to fly is a property of their manifesting. And contrary to what others have posted, the Movement power wont help unless it can already fly.
However, as Ophis points out, get to Force three and your air spirit can have the optional power of Psychokinesis. That gives it strength and quickness equal to hits on a Magic + Willpower roll, so it wont be especially impressive unless you have a sillily high force, though spend a service for the Movement power and it does get better.
If it's the flavour that you want, then I would have the character enchant the broom as a sustaining focus or for a few karma, once you are an initiate you can quicken a Levitate spell on the broom (you will want as many hits as possible on this).
Of course once you have the broom flying in this manner, then possession by a spirit with Movement power
will help you catch the golden snitch. (Sorry couldn't resist

)
Hope this helps,
-K.