I am trying to create a series of websites that are all using the same template, but the header on each is specific to the URL. How do I set this up in JMS?

Bruce..

All depends how you produce the header.
If the header is resulting of a module, you can have specific module like "customHTML" to put specific content.

If this is directly in the template (hardcoded), you can use the MULTISITES_ID to identify which is the slave site identifier and perform specific process or classes or include of other html code or whatever.

If this is just few fields that you want to customize like "COMPANY NAME", ...
You can also use the free "Search/Replace" plugin that you customize on some keyword to replace by the specific values for each websites.