Rifles and shotguns were not required to HAVE a serial number until the 1968 Gun Control Act. Many of the J C Higgins branded guns sold by Sears were made before then. And yes, it is perfectly legal...
Go to Browning.com and use the serial number look up function. You have not provided the model name, whether it is centerfire or rimfire or the caliber. Without that information, can't help you.