Considering that Jeppesen provides their products in several different formats, such as the paper charts, their JeppView application for iPad EFB's, and database subscriptions for aircraft Flight Management Systems, it seems highly likely that they use a database of their own -- and a massive one at that!
Beyond what is publically available, they offer a service to major subscribers (i.e. airlines) to "tailor" some or even all of the approach charts. While they can't change the FAA's minimums in a way that makes them less restrictive, they can do other things with them. For instance, if Sky Air only flies aircraft that are approach category C, then the tailored charts might remove the Cat A, B, and D minima. Also, if they always use the altitude alerter in the aircraft for non-precision approaches, they might have Jeppesen round the non-precision approach minima up to the next 100' increment.
I've also seen tailored charts that address Category II and Category III approach minimums. The generic chart might show both Cat IIIA and IIIB minima, but if the airline was only approved for IIIA operations, the tailored charts wouldn't have the IIIB info. Or an airline that only used autoland would omit notes about using a HUD for approaches below Cat I, while an airline that only uses a HUD and not autoland could do the reverse.
When a published change from the FAA affects a point on the chart (say, a feeder fix to the ILS course itself), then every chart with that point will have to be re-issued, and that could affect dozens or even hundreds of charts -- and attempting to ensure that process works as a manual process would be nearly impossible. A database that includes the ability for a Jeppesen charting specialist to say, "show me every chart with the point 'JEBBB' on it" would make that process a lot more reliable.
When you see some of the new products that they have in development for the iPad, such as SID and STAR charts that are to-scale and geo-referenced -- allowing the magenta aircraft symbol to "fly" along the points of the chart -- and with significant terrain screened in as a dim underlay to the graphics, it becomes clear that they are only producing these sorts of products out of a database, and not created by hand.