What was the first commercial aircraft to have a 'built-in' oxygen system?
The B307 may have been the first pressurized airliner, but it was usually flown at 12,000', and did not have a passenger supplemental oxygen system.

Boeing 307 / C-75 Stratoliner 
1938

"Captain Frank E. Adams, 23.02.2011
I flew the S-307 2 years for Pan American Airways (1943/1944) and today remain one of the few pilots in the world who is still rated for this aircraft. My biggest complaint about this description is that the aircraft could only be pressurized to a maximum of 2.4 psi so that the advertized service ceiling of 26,200 feet is greatly exagerated. A flight altitude of 22,000 feet would result in a cabin altitude of 14,000 feet and we had no passenger emergency oxygen on board. Also,the Wright engines had no turbos and so were basically sea-level engines which could not retain cruising power output above 12,000 feet. For this reason we flew the S-307 at 12,000 feet southbound and 11,500 feet northbound between Miami and Panama."

Later pressurized airliners like the Douglas DC-7, Lockheed Constellation and De Havilland Comet were flown much higher and I expect they would have had some form of built-in supplemental oxygen system for the passengers. (unfortunately I can't find any evidence online)

I did find an advertisement from January 1957 which shows a Scott passenger oxygen system which would represent what was used on the Boeing 707. (first entered service in October 1958)

Flying Magazine Jan 1957