Different states have different requirements. A few require background checks, some more require a license dependent on what type of weapon it's for, quite a few have bans on AP ammo, but you can have ammo delivered to your home without any restriction in more states than not. Also buying ammo has an age requirement of 21 in some states in case you're under that.
Specifically for shipping to a motel, I would say they may refuse accepting the package if it is marked by the company you're selling it making it is obviously firearm related, though I've generally found companies ship without logos on the box. If it's rounds/small arms ammo though, it has to be marked as such on the package, but not an issue if you're ordering shotgun shells. Never worked in a motel, so I may be just assuming they would have a problem with it but I imagine most motels probably have policies against having firearms in their rooms so last thing you want is them contacting the police and finding it inside.