Although the Schwinn online store does not carry parts, there are several other online stores who do. These range from major retailers such as Amazon.com and eBay to more specialized outlets.
Amazon.com and eBay have extremely large selections of bicycle parts as of 2015, including new, used and vintage parts. If you're looking for a retailer that caters specifically to bicycles, check Bicycle Heaven, which claims to be the "Worlds Largest Bicycle Museum and Shop." Schwinn Cruisers is a website that focuses on vintage Schwinn parts in particular. Vintage Schwinn carries a number of rare parts that can't be found anywhere else.