This is not a conclusive answer, but it's a starting point to look at patents on the particular devices.

There are several different patents for starting devices, most of them were patented from 1910 to 1920.

Charles Kettering Patented a electric "engine starting device" with a priority date of April 17th, 1911 but it wasn't activated by a key, rather it was activated by a foot pedal. Charles Kettering was an engineer with Dayton Engineering Laboratories Company (DELCO). You;d recognize them today as half of AC DELCO, which is a subsidiary of GM.

Kettering likewise recieved a patent on a dashboard mounted starter - the drawings show sort of a toggle switch or lever. He applied for that patent on March 12, 1910 and recieved it on June 26, 1917.

About the same time Vincent Bendix invented the Bendix Starter which had patent priority of March 15, 1916. Although it's an automative website, you can see a detailed description of the part here.

Lots of websites suggest that Bendix also patented a turn-key ignition in 1910, which was then placed into early model cadillacs, but I can't find any patent sources for that.

Something which many people may not realize is how starting a car works. A hand cranked starting car works much like a hand pulled lawnmower. You use muscle power to turn over the cylinders and create the compression necessary to ignite the gas.

Cars use an electric starter motor, which is a small secondary motor to accomplish this purpose. This motor requires a battery. If this motor is running, it runs down the battery. So for cars with electric starter motors, you need a switch to (A) turn on the starter motor, and then (B) trigger the starter motor to start the engine.

Although I can't find a patent directly tied to using a key for ignition, this popular mechanics from 1949 lists it as a chrysler innovation that there is a combined starter and ignition swtich, removing the need for a starter button. Again, that's not conclusive as to when it became an industry standard feature, but it wasn't long after that that it became ubiquitous to have a single key that would both "turn on" the car and turn to ignite the engine.

Edit: fixed the last sentence.