The Fullscreen API is not supported in iOS Safari:
http://caniuse.com/#feat=fullscreen
And since Chrome for iOS is based on iOS Safari (rather than Blink), it too does not support it.
In iOS Safari, the browser does go fullscreen if you switch to landscape (say for Cardboard VR rendering), but iOS Chrome can’t do this.
According to webglstats.com, iOS Chrome seems to have about 5% of iOS traffic to the sites listed. Slightly more than I expected, TBH.
You could try opening a bug on Chromium if you’d like Google to look into this.