I've seen this a few times. It's usually not quite as insane as it looks. The underlying reasons that I've seen are:
* The web site is tied to an old fashioned batch system/mainframe that doesn't run in the same mode 24/7. Yes, the implementors could write a separate queue that managed this - but that would be often be a non-trivial amount of work.
* It's for a system that provides the online equivalent of an offline process, which has escapes that occasionally push the user out to talk to a human being. The human's aren't available 24/7.
* It's for a system that provides the online equivalent of an offline process, and there are legal or social requirements that the applications be "fair" - i.e. that the people going through the online and offline process should have the same opportunities. Having the online system run 24/7 puts people with online access at a significant advantage.
Could be an ADA issue... ADA (or the state's interpretation/augmentation of it) may say that they need to have live assistance available and they don't want to staff that 24/7 (this is a business registration site, most usage probably happens during business hours).
I did a small amount of contracting for a local real estate company who only had their site online certain hours of the day. I asked them why, and their response was that they only had the one server, and during certain hours they needed it for processing new listings and database work. This was apparently such an intensive task on their budget server that the web server became unresponsive during these operations.
They also said their analytics showed that no one ever visited the site outside of business hours. This wasn't what I was contracted to do (and I wasn't in position to fix it), so I didn't bother to tell them that when they shut down the web server their analytics stopped running as well.
Also, if you have no or limited resources with which to handle off-hours system issues (errors, intrusions, etc), unplugging the site can be one form of KISS. (Assuming you know your audience will not migrate away or vote you out of office, etc.)
If you (effectively) subdivide your content pages from your processing pages (on a separate system), you can simply take the latter down.
It may not be ideal for today's idea of a 24/7 world. But if you're small, geographically local, and want your staff to have sane lives, it's possibly one approach to consider.
(And there are currently arguments circulating for reinserting some sanity and balance into the way we live our lives and jobs.)