I noticed that Chromecast has an iOS SDK, that seems to allow similar behaviour to airplay in a iOS app. Does Roku provide a similar ability? I can't seem to locate anything.
Yes and no. It's complicated.
Roku supports Miracast, so if your app or device has it, it can "project" to Roku. This is easily done from Windows 8+ (PC or phone/tablet) or Android.
Since you said iOS though, it does not support Miracast (which requires some low-level wifi things, can't be done purely by library). Nor is there video AirPlay outside Apple (ask Apple why). So i can't think of a way.
Roku supports DIAL though, that's how videos can be cast from mobile Netflix and YouTube to a Roku.