OK... Just to get some idea of what I "might" be up against...

[Dreambird]

Congrats Raven, I'm happy to hear that you have things working better now! 
When I got the current MacBook Pro I'm using right now, it didn't have firewire ports so I did have to use an adapter so I could use a thunderbolt port for firewire. Anticipating that I purchase the adapter I needed with the machine so setting up the new machine was a breeze with Migration Assistant.

In any case I understand that you just want another year or two to run what you've got without problems, that's all I really wanted too and so I hoped to upgrade to High Sierra and get that time but so far still no luck. I haven't tried the command line in Terminal yet, I hope it might work but apparently it doesn't always. Another trick might be to download Chrome browser since it seems to "fetch" when Safari and Firefox don't. I did ask my sister who runs Big Sur to try and get High Sierra from the App Store but that got nothing as the screen shot I'll attach shows. I'll have to come back in Safari to do that since the old Firefox I have won't. They've dropped support for it as of El Capitan but I still get security updates for awhile.

[Dreambird]

My sister could not find High Sierra or even El Capitan on her machine running Big Sur.

[Xairbusdriver]

This link takes you to an Apple Support (US) page. However, the links to download six different older versions of the Mac OS (Catalina, Mojave, High Sierra, Sierra, El Capitan & Yosemite) take you to the (hopefully CA) App Store. Did you actually use the following link and then go to the App Store?

https://support.apple.com/en-us/HT211683

[Dreambird]

Yes, I have tried that link using Safari as the info on the link instructs, I've also tried using Firefox which first opens iTunes and there I get a message that the item is not available in the Canadian store. When I quit iTunes because obviously that's not where to find it anyway. Go back to Firefox I find that it gives the same error message as I get in Safari.

It simply states that in order to use the App Store I must have OS 10.6.6.
I have OS 10.11.6 which is El Capitan, which it does not seem to recognize. But that IS what I have... under "About this Mac" it even clearly states OS X El Capitan, Version 10.11.6. It doesn't seem that the "us" in the URL is the problem but I have tried changing it to "ca."

I do in fact have a full installer of High Sierra that I downloaded from the App Store before it was taken off there when the new OS came out. I won't know until I try it if it actually works because apparently Apple might not have signed it. I even looked in my purchased list in the App Store and all that shows there is El Capitan. I think before I risk ruining what I have right now on my MacBook Pro is that I might try using the HS installer I do have on an external drive just to hopefully find out if it will work to upgrade a clone of El Capitan there.

[Paddy]

Very odd. When I click on the download link for High Sierra in any browser (Safari, Chrome etc.) I get that same "Connecting to Mac App Store" message - but then the App Store OPENS - and it's the Canadian one (the help docs, Jim, are always US-en, but Apple figures things out when you connect to the App Store and opens the Canadian one). I'm running Mojave on this particular machine - don't have anything older now.

So it's not the message - it's the fact that the App Store isn't opening. However, there is/was a message asking if I wanted to go to the App Store when I first clicked on the link. I said yes - whereupon it opened every other time. Perhaps you told it "no" at some point. Try dumping your cookies - that preference is likely set by the browser. Worth a shot. (Couldn't find anything from Googling this issue)

[Xairbusdriver]

I'm not sure the OS Dreambird is using has the "App Store". That may be why she is getting "iTunes" instead of "App Store".

...I have tried that link using Safari as the info on the link instructs, I've also tried using Firefox which first opens iTunes [emphasis added]

If you want to be sure your browser is not using its cached files, you should use its 'empty cache' item. In Safari, that function is not available until one enables the "Develop" menu. The enabling preference is in the General tab of Safari Prefs, at least in recent versions. I have no idea where it is in any of her OS's.

All this is beginning to look like the infamous "Chicken or Egg" problem. I have no other suggestions than finding a friend who can get and then put the desired installer on a thumb drive. Sometimes 'sneaker-net' is the easiest way to move files.

[Dreambird]

Thank you Paddy and Jim... I will try your suggestions to clear caches etc. I do have the Develop menu in my Safari. I don't recall ever being offered the option to choose the App Store, however the first time I tried the download with my Safari it DID "start" the download but quit after about 2min. and from then on I get only the error message about the OS version. I used the instructions I found in a thread re: Raven's problem getting Mojave to make sure any partial download was gone.

Yes, Firefox will open iTunes looking for the download but when I quit that, back on the "desktop" I use for Firefox is the OS version error. My Firefox is still up to date so far as security updates go but that will run out soon as they've stopped support for anything less than OS 10.12.x
Firefox is somewhat notorious for not playing well with some sites too though. I've not had a lot of trouble, but this might be the one that is an issue. If I manage to upgrade I may download Chrome as that one seems more reliable.

My sister was not able to turn up a High Sierra download on her iMac running Big Sur but I could still send her the link to Apple's download site and she if she has any success. She runs both Safari and Chrome.