Air carrier Captain reported refusing an aircraft due to APU inoperative; high cabin temperature; and passengers becoming ill due to the heat.

Narrative

Assigned aircraft with an APU bleed air INOP. Upon arrival at the gate the aircraft was reasonably cool. Once the passengers began to load along with the increase in outside air temperature the aircraft began to become increasingly uncomfortable. At approximately 25 min prior to push; Operations was called to inquire about the location of the external air cart. Calls were also made about every 10 minutes as to the location of the cart. At 15 min after push the aircraft temperature had become uncomfortable. At 25 min after push time the cart arrived. I was unable to get the ground crew to start the air to the jet so we could run the packs. Standard comm procedure was not followed by the ground crew throughout the air start. Quote from Operations at 15 after push we knew about the INOP APU prior to this bank. It's on its way from [Gate] XX"We finally got ENG 1 started and pushed back. Followed closely by a start of ENG 2 via cross bleed. Aircraft cabin temp was showing over 30c but cooling. On taxi out FA (Flight Attendant) calls to inform that the pax in 2A is having a hard time breathing and is vomiting. We immediately inform ground and begin a return to the gate at approximately XA:25. When we block back in paramedics are waiting. I asked Operations to have the air cart brought over so it could meet us and get cooling air into the aircraft. Gate air was pumping in at around 28c with multiple kinks in the hose. Air cart arrived but was never hooked up. Aircraft temp continued to rise. Approximately 42c on the aft and 38c in the front. At this time I made the decision to deplane the aircraft and file the Flight Did Not Operate. Called Dispatched and conferenced with Maintenance Control. Both agreed and Dispatch entered the code. Once the incapacitated passenger was cleared from the jet bridge the aircraft was deplaned."