We don't control how platform makers deal with photos on their phone's browsers. So, this isn't our decision to make. Apple will release a new iOS version this fall which will allow for the uploading of photos from iPhones.
Are there any options available for me to be able to create a native iPhone app that allows my members to post photos they have snapped with their camera to their photos page?
Does the Ning API allow this? Like could I hire Shoutem to create this for me?
Sure. You can absolutely hire someone to add this capability. Go for it. Or, you can wait for Apple's iOS update, which will be free and will be out this fall.