Connecting Flights on Separate Tickets
 

We are traveling from Chicago to Madrid (with a layover in Lisbon). Once we are in Madrid, we have a long layover before continuing to Las Palmas, Spain on a different airline. We are not checking any luggage. We will check-in for both flights before we leave home. When we are in Madrid, do we need to go out through security to the check-in desk for the different airline or can we go straight to the gate and let them know we have arrived?

Depends if they'll issue you a boarding pass or not when you check-in online for that second airline. If you get that you can proceed to the gate and no reason to announce yourself. If not you can try at the Transfer Desk.

Just to be clear you don't have a connection in MAD as far as the airlines are concerned. If you are delayed getting to MAD on separate tickets neither airline is responsible for getting you to Las Palmas so I hope you have a good amount of time between flights. Upside is that you'll clear Schengen immigration at LIS and your flights in MAD will arrive as a "domestic" flight.