So I'm pretty sure this is not possible, but would love to get some validation. I'm interested in prompting the user to save an image to our app or perhaps be given the option to be directed to our app after an image is taken via Apple's native camera app. This presumably would require some approval/permission from the user, assuming this sort of thing is allowed by Apple to begin with. Any thoughts/suggestion would be greatly appreciated.
This is not possible without a jailbroken device.
Apps typically run in a sandboxed environment, unless jailbroken. To hook into the Camera app, which is essentially what you are looking to do, you would need Mobile Substrate and some crafty code to get in there.
The only way for your app to save photos in itself (I'm assuming the Documents directory or some local database?), your app will need to take the photo.
You can always have your app create a Photo Album (simple enough), and if the user wants to categorize their photos they can do it that way, but you would need to prompt the user from YOUR app when they open it.
You CAN however, access the Photo Library from your app and import photos the user has taken, but this would again require user interaction, and they MUST allow your app permission to access the Camera Roll, or Photo Library.