Never mind. I figured it out. Here's how. I don't know if any of these steps are unnecessary but it's what I did.
First, go to the Met home page & hit F12 in Firefox. It is important to have the monitor open BEFORE you visit the actual stream page. Things you want to observe appear when you first open the page, & visiting the stream page first & then hitting F12 will miss some things. After you've got the monitor open, then click the link that takes you to the page where the stream is located.
Actually, come to think of it, given what I'm about to say, it may not be so important to already have the monitor open. But it's what I did. In any case, when you first visit the stream page, before you even launch the stream, the monitor will show you that you have received 3 manifest files. (See my attached image Fille#1.) You can inspect anything in the monitor by just double clicking on it. Firefox prompted me for what to do with the resulting file so I told it to open it in my Windows 7 text editor. (I'm actually using a Notepad replacement but Notepad is fine. Pick your favorite text editor.) The first time I did this, Firefox was completely at a loss how to deal with this file so I went through the Windows dialog for supplying some application other than Firefox to handle the file. This became a sticky setting & I don't have to go through the dialog every time, just tell it yes use the text editor again like last time.
Each of these manifest files contains a number of URLs. Each of those URLs points to another manifest file. I clicked through all the URLs in all 3 of these manifests & did NOT find one for the captions. So I went ahead & launched playback. Then I enabled the captions within the stream as it was playing.
Next, I looked in the monitor again. Of course, bunches of new entries had been added because now I was receiving chunks of the stream. I scrolled down through the monitor until I found another entry for a manifest file. It's also identified as x-mpegurl in the Type column. Hovering the mouse over that gives you the tooltip application/x-mpegurl as shown in my image Fille#1. Double clicking through that let me inspect this later manifest. I'm showing the first few lines of this later manifest in my image Fille#2. This is the manifest we want. The caption type being used in this case happens to be VTT.
Once you've discovered the manifest for the captions, you need to get the URL of the manifest. Go back to the monitor & pop up the context menu on that entry by single clicking mouse button 2. (See attached image Fille#3.) Then select the Copy URL option. This puts the URL of the manifest into your clipboard. Next, paste that URL into a plain text file using your text editor. You don't want to lose this URL so put it somewhere safe.
After asking my question last night, which I see now is a bit silly, I thought I might go look up the documentation on Ffmpeg. (I wrote all of this before I saw that MgFrobozz had so graciously provided the above information.) I read a bunch of bewildering stuff but I came away with just barely enough to do what I needed in this case. We all have Ffmpeg because it's packaged with VDH. To be more accurate, I believe it's actually packaged with the CoApp. But since we are all using the CoApp (we are all using the CoApp, right?), we all have Ffmpeg. The only tricky bit is finding it. On my system, it's buried under Program Files on my boot partition. Use a dir /s in a command window to find it. The executable name is, unsurprisingly, ffmpeg.exe. Once you've found it, copy/paste that file spec into the same file in front of the URL for the manifest that you saved above. You're constructing a .bat file here, in case you were wondering where I'm going with this.
So the first (and only) line of our .bat file invokes Ffmpeg. Since the executable is within a directory tree that includes Program Files, there's a big fat space in the middle of the path to the file. So you're going to have to surround that part of the line with quotes. The first argument in the invocation of Ffmpeg is -i. The -i argument tells Ffmpeg where to get its input. Its input is the manifest for the captions we pasted in before.
The last part of the puzzle is specifying the file on your system where you want Ffmpeg to place the captions. The entire command looks like this:
ffmpeg -i URL targetfilename
On my system, it looks like this:
"Q:\Opera\La Fille du Régiment Video 20080426.vtt"
The quotation marks are critical. This won't work without them because of the spaces in the various file names. Also, this is actually a single line. I'm showing it folded here so you can see all the parts. The parts are separated by a single space, not a line break. I've shortened the URL to the manifest because it's long. Plus the name you will see will be different, especially in the part after the question mark.
Now save this arcane load if gibberish into a single-line .bat file named whatever you please. Then execute it. A bunch of messages will flash by in the command window. You can execute the .bat file by just double clicking it. If you want to see the messages & read them at your leisure, first open a command window & execute your .bat file from there. The .vtt caption file that Ffmpeg created for this opera was barely 52K. Tiny. It took only a few seconds for Ffmpeg to download it.
In an excess of snobbery, I have named my target file with an accented character. This matches the name of the .mp4 file of the video stream I saved earlier. The matching of the .vtt with the .mp4 satisfies a feature of VLC, as I mentioned above. Ffmpeg did have a bit of trouble with the accented character. The file name I got included what looked like a Greek capital theta in that position. After the fact, I just renamed the file to put the accented e back in the file name & all was well after that.
As I suspected, I did not need to explicitly tell VLC about the caption file. I just needed to tell VLC where the synchronous video & audio files are. The simple existence of the .vtt file in the same directory with the .mp4 file got the captions to display. Of course, it isn't until after Renée's introduction & after the overture that you actually see any captions. But they do display.
One possible bonus of having the captions in a separate file like this is you can correct them if the urge takes you. I doubt it will take me but it's possible there's a word here or there that might be misspelled. For any sticklers, if you happen to understand the language of the opera, you might want to translate some line with different words. That's easy to do. Just edit the .vtt file like any other plain text file. If you're going to do any extensive work on the captions, I would recommend getting a caption editing application. You can search for them on Google. One I have used in another context is called SubTitle Edit. It's free & it is being actively developed.
Now, I gather Ffmpeg can also combine the video, audio, & captions into a single file. I don't really care to do that. I can play this back whenever I choose & I will see the action, hear the singing, & read the captions just fine with the 3 separate files.
This process should generalize to any future opera streams the Met will put up. Since URLs will change, the instructions will change as well. Just edit the .bat file to indicate the new URL & the new target file.
I hope this helps somebody. And thanks to MgFrobozz for your participation. Evidently you're running Linux. I'm on Windows. It's good to have all this variety.