Unfortunately for those who fly Delta, Delta is not partners with Iberia, so getting a ticket from Santiago to US on Delta or with a Delta code share will not be easy. I THINK that the best bet for Delta passengers who want to fly from Santiago is to get yourself to A Coruña or Vigo, where some Delta partners fly. You can then get a code share ticket.
My guess is that this is two separate tickets (I think Vueling only has a code share with Iberia and One World Alliance), but maybe you have been able to book it all on one ticket. Especially for the trip home to the US from Europe, I think most of us would strongly recommend that you have one ticket for your flights. If the Vueling flight is cancelled or has a huge delay, and you miss your connection in Amsterdam, you will most likely have to buy a new one-way ticket home. That would be very expensive.
I am hoping that
@BillW was able to find a way to book this all on one ticket — if not, I think the best options are train/bus to Madrid or a flight to some European city from either Vigo or A Coruña and then a transfer to your US-bound flight.