We have a 12 year old ICF basement and it is leaking at the bottom of the wall where it meets the floor. A contractor in my area advised me to pull of some of the styrofoam to try and see if it is leaking down from a window or where else it is coming from. When I did this it is only wet at the bottom and I have reached sand/dirt below the styrofoam and not a footing. Everything I have read says the footings need to be wider than the walls, that the blocks should never hang over. Any suggestions? Please help.

Go back to the building department and get a copy of the original stamped plans which the house was built from. Your footing maybe further down and not necessarily right below the floor.

So, it's a full depth basement with no footings; just the ICF blocks set on undisturbed soil, or does the ICF just hang over the footing at this one point?

No gravel or self-compacting rock under the floor or the ICF walls?

Full height backfill all the way around outside the basement?

It could also be built as a grade beam on piles, rather than on a footing. Did you see any type of frost cushion (typically cardboard or styrofoam product)?

If you have strong enough soil (say around 4000 psf), then structurally, you don't need a footing.

Jonr is right, from a theoretical standpoint. And under a lot of conditions a concrete wall will perform OK without one. Lots of discussion can be made there. That aside, you'll need to determine **precisely** what the issue is, then it can be addressed. A few things you'll want to check are:

Is this a locallized problem rather than consistent around?
Has any vertical movement taken place?
Is there a crack where the water is coming in?
What is the exact path of water intrusion?
How much water, and is it seasonal?

Other questions will arise as you go along. Once you have enough answers, then you can determine a reasonable course of action, depending on what you can "live with" and what price tag the ideal solution comes to.