Wikipedia's article leads to several sources that substantiate what @ValentinGregoire preempted. For example, Chimay put up a series of short video clips that take us through surprisingly up-to-date facilities capable of bottling up to 40,000 bottles a day! (Probably it's just my own prejudices—associating monks and monasteries with old times, thus old technologies—causing surprise in my case.) With many other competing beers, and demand checked by relatively higher prices, the abbey alone suffices.
The volume produced, according to this 2012 article, is 120,000 hectoliters of beer annually, or 16,000,000 bottles (750 mL) of beer. To put this in some perspective, according to this 2008 article St. Bernardus at the time was exporting to 20 countries and producing 13,000 hectoliters (considerably lesser than Chimay) annually. And as @ValentinGregoire started saying, Westvleteren is difficult to find because the monks of the abbey of Saint Sixtus decided not to increase production despite the beer's popularity, and thus produce only 4,800 hectoliters according to a 2005 publication.
For me, it's hard to grasp such large numbers in any meaningful way. I looked up Stella Artois' production figures (and not because it's Belgian—it was just the first popular, global beer to come to mind), and found that in 2012 they produced a little over 10,000,000 hectoliters that year. Divide that by Chimay's 120,000 hectoliters per annum. Can I believe that 83 units of Stella are being consumed for every 1 equivalent unit of Chimay? Meant only as a sanity check and not any rigorous argument, it checks out as believable to me—after all, Stella's on draught all over the U.S. (I can't speak for other countries but I'm sure everywhere else too).