Yes you can! I’ve delivered many a bicycle at my day job driving for UPS. FedEx also offers this service.
The biggest pain for you shipping a bike is going to be the packaging. If you’re doing it yourself I suggest measuring the dimensions of your bike and getting a large box. The cardboard box you’ll be looking for is a “large side loader”.
ULINE has some available: https://www.uline.com/BL_413/Large-Side-Loaders
My biggest gripe with these boxes is that people are usually shipping a decently expensive bicycle in them. But not spending the time to pack them correctly. If you’re going to pay a good chunk of change to ship and package your bicycle – please consider going all the way and prevent the bike from shifting freely in the box.
Just imagine Shrek was the one loading your bike and he has had a few too many. With that in mind – you can pack accordingly!