This is actually not very true despite it commonly being said.
Most state laws only specify bans on animals like native/exotic venomous reptiles, bears, big cats, and large canids or "inherently dangerous" species. A few states have
no restrictions at all.. Most other native species (opossums, squirrels, raptors, etc.) aren't covered. A few states specify raccoons as either being illegal or requiring a permit. Actually, relatively few states actually outright say
anything about native wildlife species in these laws other than bears/wolves/alligators/big cats. I had to sift through all of these laws a few months ago for a project, fun times.
You also do not really need a rehabber permit specifically. If a state requires a permit, you can apply for a permit just to own the animal (if required by your state). Most people are given the permit. If you are in a state where you need a permit and don't have one, they likely still wouldn't even confiscate the animal tbh. If it's not posing a threat to the public, getting loose and running around a neighborhood, whatever, no one is going to bother.
Fun fact, but in Alabama, it's illegal to have a raccoon but completely legal to have a lion (with no permit, might I add).
Summary of State Laws Relating to Private Possession of Exotic Animals is a decent source, their summary descriptions of the laws for each state are accurate.
ETA: Also want to mention that nearly any of the wildlife laws can be surpassed with a loophole. Maybe you can't have a raccoon as a pet, but you can harbor a raccoon with the intent to breed it for game/fur or harvest its fur (and with a trapping/fur license) in the state of Illinois. People get clever with this stuff, some people even go as far as hiring lawyers to interpret the laws and find them loopholes so they can get these animals.