Closing in on midnight here, so I'm going to hit the hay and check back tomorrow morning.
Thank you for your responses!
The clients I mentioned attach to your TV. You add to Channels DVR app to them. Watch a YouTube video that compares them and the various Android TV options
Every media player is going to have a different way to select closed captions.
For VLC you select them from the Subtitle menu

or right click in the video

Channels clients Channels — Apps for TV, Tablets, and Phones
The web player isn't a proper client. It is basically for diagnostic purposes and not something you should rely on for viewing your content. You'd be better served by getting a client like Android tv (Walmart Onn devices are really cheap and do quite well) or appleTV which will give the best experience.
But I don't have a "TV". And if I did it sure as heck wouldn't be a 'smart' one. 
You could install an Android emulator on your PC and then install the Channels Android client in the emulator.
Okay, still fuzzy on the whole client concept, but isn't the web player something that's used for live TV? I just want to get closed captions when I use a standard media player to play the mpg files that Channels has recorded for me.
The web player is not a full client yet you can watch content using it. The developers have been asked to make it a full client yet they have stated that they are not going to do this. I don't recall a reason.
Thanks! I think I previously tried the standard approaches for turning on subtitles/closed captions in MPC and VLC, but I'll give it another shot. I wanted to make sure there wasn't some setting in Channels that needed to be a certain way in order for the sub track to be recorded in the first place.
Channels DVR records whatever the source sends, so if it sends closed captions they will be recorded in the file.
Pretty sure these Windows apps also support closed captions.
Media Player
Movies & TV
There is also an optional integration that allows you to play live tv and recordings from the Channels DVR web UI using VLC. Settings > Advanced > Integrations > VLC links
Show VLC links in the web UI. Requires installation of vlc-protocol helpers.
It will be interesting to see how Channels DVR handles Closed Captions as things evolve over the next year or three as ATSC3.0/DRM takes over.
Take a look here
Pick up a supported client and connect it to your TV via HDMI. Closed captions will work beautifully and you will have the best experience. Drop the web player idea, you are asking for nothing but frustration. Sure there are hacks to get it to work with VLC but the easiest thing to do is get a supported client and sit back and enjoy your content
Think you missed his earlier post
You could buy a supported client and attach it to an HDMI switch attached to a PC monitor, but why?
Ahh gotcha sorry. Better add a TV to the shopping list as well lol
That is unless his monitor has a free HDMI port and speakers. Then he can use it as a TV
I think you meant monitor, not TV?
I watch most of my Live TV and recordings using a monitor, except when remote and use the iOS app.
Thank you once again chDVRuser for seeming to 'read the room' correctly in my case.
It sounds like there shouldn't be a problem with most standard television broadcasting having the potential for closed captions, so my DVR recordings ought to be able to show them. I guess I'm just going to have to figure out whatever I'm overlooking. Probably something stupid!
(For what it's worth, if anyone cares, I'm an old fart Boomer, and have no desire for any 'smart' devices whatsoever. Those of you who have no problem with a live microphone in the middle of your living room 24 hours a day that could theoretically broadcast to anywhere in the world are fine by me, but please don't try to get me to jump on that bandwagon. I don't even like hearing the word 'Android', because I dislike telephones in general, and only use my completely 'app-less' one to occasionally make a phone call or get a voicemail. My computer is hard-wired to my internet provider and has no camera or wifi, since I just don't have any need for this sort of stuff.)
Sorry to be boring, but I just felt a need to come clean! 
If you install the channels app on your phone you will have solved your problem as things will work as they are designed
Like the OP, I prefer to watch on a larger screen (a Monitor) and only use my iPhone/iPad when remote.
I sometimes use an HDMI adapter output from my iPad to the TV HDMI input when in hotels.
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