The house is new, as in new construction? Or is it old construction, but a new purchase?
If the former, you can try contacting the builder. If the island was properly plumbed, you may simply need a new vent, which is the thing that is screwed on to the top of the pipe that you found under the sink. In my area the ones that are carried are Studor Vents.
Studor Vents have a small rubber diaphragm that can flex. Normally is seal the pipe, but when you use the sink and create a siphon, the diaphragm flexes and allows room air to be sucked into the pipe, breaking the siphon. Think of it as a one-way check valve for air. Air can be pulled through the valve and down the pipe. Air should no be able to come out of the pipe and into the house.
1) The Studor Vent could be bad. It may not be resealing, and it's allowing septic odors into the house, thus the smell from the vent.
2) The Studor Vent could be bad. It may not be flexing at all, so it's allowing a siphon to occur. When you use the island sink water is being sucked out of the P-trap, breaking the seal. Again, that will allow odors to enter the house, but they'll come from the drain in the sink.
3) The island may be plumbed improperly. I've seen many a plumber screw up many an island sink. It's easy to do properly, but its easy to screw up as well.
As an aside, if your septic is new, it will smell for several months until the tank gets into balance. Assuming you have a tank.
First call? The builder. The smell is making you sick, right? That's what you want to tell him. I'm sure you're sick of the smell. And he should send someone ASAP, as you fear you may need to move out of the house and into a hotel until the problem is solved. With the hotel billable to the builder, of course. Ask him what hotel he would like you to stay at.