A conjunction or preposition between the two system-names is needed to make the name grammatically sound; for example, both of "Facebook and Flickr Connector" and "Facebook to Flickr Connector" are correct and understandable. "Facebook/Flickr Connector" also is a possibility. [Note, company names randomly chosen.]
By contrast, "Facebook Flickr Connector" looks like two adjectives and a noun, that is, would represent a connector for the "Facebook version of Flickr", if there were such a thing.
Your alternate terms, gateway and exchanger, are reasonable alternatives to connector, except that connector is a term frequently used in enterprise application integration, so may be a good fit with your Enterprise Resource Planning to online shopping synchronizer.