As long as you are all sharing the same room, you should have no problem finding a room within your $120 a night budget. Most places, you can find a budget to mid-level room for $50-70, so that would even allow you to occationally get a break by having two rooms.

If you are staying right within, or just outside, major national parks like the Grand Canyon then you'll likely need to spend more than $100, but even there, there are typically more budget friendly places if you are willing to drive. For example, for the Grand Canyon, you'll find cheap lodging available in places like Williams or Flagstaff an hour away.