How are aviation sectionals and enroute charts created?
I believe the FAA got out of the printing business by implementing a new business model for charts where they license the product to private industry who then take on the risk/reward of the printing process for their customers.

From the FAA AIS website: 


  The Federal Aviation Administration (FAA), Aeronautical Information Services (AJV-5), is transitioning aeronautical charts and related products print and distribution operations from existing contract(s) to multiple no-cost agreements with private industry. Digital product files allow industry the opportunity to print, sell, and distribute aeronautical charting products to the public in an Available on Demand (AOD) approach. Any party interested in the opportunity to become an approved print provider should submit product qualification samples from the files below (in accordance with instructions in the Print Provider agreement).


More detailed information about the program is also available through the FAA contract award management site via an official announcement. So to answer your question, the printing of charts is now handled by private industry, not the FAA.

Since you mentioned Jeppesen specifically, note that they are a bit of a different concept because they take raw data from each state's official AIP and then integrate it together in their own way and perform other value-added services. They then print their own end-product for their customers.  It is based on the same raw data but their product should not be considered the same as what is produced by the FAA.