I don't think you would be able to incubate wild birds for release. As stated earlier, the only birds that aren't federally protected are the pests, such as the sparrow, pigeons, and starling. And we don't need anymore of those released since they are a non-native pest.
I think the ONLY way to do this legally is to become part of a conservation or special research project. Then again, I'm sure that these aren't easy to get into and have strict limitations. I know that some endangered raptors went through a conservation program, in which scientists and whatnot would take all but one egg, replace them with wooden eggs and then incubate the collected eggs. This way the endangered parents could give the remaining egg all the attention and increase the survival of that one chick, while it's siblings were raised for either captive breeding or release.
This isn't just a matter of obtaining a wildlife rehabber permit, but you would have to get federal permits and probably become part of a special team or effort.
If you are wanting to help repopulate native species, the best thing to do is set up species specific houses. Provide plenty of food sources, seed and plants and maybe even mealworms. Then cut back on all the non-native species that are threatening native bird populations. This means trapping feral cats and taking them to the humane society, trapping and killing invasive species of sparrow and starlings.
If you google "starling traps" or "sparrow traps" and even "pigeon traps" you will find traps you can make to capture these birds. I read that once the birds killed, you can donate frozen starling and sparrow and probably pigeon bodies to a local raptor rescue. These traps are used by big time wild bird enthusiast and are typically live traps, that way any protected non-target species can be easily released, unharmed.
We have bluebird houses that we maintain on our property. Cleaned out yearly, and if they are ever inhabitated by sparrows or starlings, they are evicted immediately.
Populations will bounce back on their own, if the means are provided. Between food, housing, and keeping non-native preds(feral cats) and competitors(starling/sparrow/pigeon) down, you will help your local bird population significantly and probably stay pretty busy!
-Kim