You have enough room. You may need to muck around a bit to get a planting hole big enough due to the roots. If you need to because of the oak roots, I would suggest you plant the aborvitaes a little shallower and bring in some soil to raise the surrounding grade. Don't do this more than about 4".
Most arborvitaes are not very shade tolerant, I'm assuming the oak canopies are high enough that they won't totally shade out the arborvitaes.