iPhones don't allow the user to select the audio output destination by default
While iOS devices have so far not allowed the user to select the audio output destination of VDO.Ninja via Safari, there are some experimental options in the Advanced settings menu of Safari.
You can access such experimental option via Settings -> Safari ->Advanced -> Allow speaker device selection, as seen in the below screen capture.
There is no guarantee this will work, and maybe it will be enabled by default in the future.
There are some other experimental options as well that you can enable on iOS, such as AV1 and H265 codec support, and perhaps in the future Screen Capture will work, too. iPadOS tends to have more functionality available, such as external USB video device support (as of iPadOS 17), as least compared to the iOS 16 at time of writing.