You pretty much answer the question yourselves. Only Chrome, Firefox and Opera (desktop from next version) currently support getUserMedia
.
As Flash is unavailable on iOS (as you also mention) you would need to write a native application to support that OS.
So in short: there does not exist a single cross-browser solution for this at the time of this writing.
There are many solutions out there which allow you to grab a video from a web camera running in native environment. You don't mention programming platform so I can't give any specific example but for iOS this can be a good starting point.