If you visit those sites, does their favicon appear? (please note that not all sites have favicons)

No, they don't! Instead of the favicon, which I suppose to be the icon that appears in the address field right before https, there is a grey lock image.

I suppose that means it's either Firefox's problem or the web site's problem, right?

Please create a screenshot of the problem you're seeing by following the guide mentioned at How do I create a screenshot of my problem?.

Once you've done this, attach the saved screenshot file to your forum post by clicking the Browse... button below the Post your reply box. This will help us to visualize the problem.

Here is the screen shot for the web site of my medical provider. It shows the bookmarks bar, with the bookmark PAMF-BG I used to access the web site.

It's possible that the bookmark you have in your bookmark bar is pointing to a page that then redirects you to another page. The redirecting page doesn't have a favicon, but the page it points to does. Hover over the bookmark, and see what website it points too (should appear in the lower left hand corner of Firefox. If it is different than the actual website address you end up on that is the reason.

The web page in your screenshot doesn't show a blue highlighted star on the location bar to indicate that the page has been bookmarked.
You can check the URL of a bookmark via the properties of that bookmark (right-click context menu) and in the Bookmarks Manager (Library)

• Bookmarks > Show All Bookmarks

The bookmark has the following URL:

https://mychart.sutterhealth.org/?mode=home&aff=2&onepasswdfill=8DE41DBBBFDD492B9A18CC03A45EE433

Now this URL comes from the 1Password system and goes to the home page of Sutter Health, which is

https://myhealthonline.sutterhealth.org/mho/?mode=home&aff=26

It logs me in, and then the page I got a screen shot for is displayed.

But the issue here is that the bookmark on my bookmarks toolbar has no icon, but rather an empty square. I am asking is there is a way I can get the icon of the home page to show there.

See:

• Bookmark Favicon Changer: https://addons.mozilla.org/firefox/addon/bookmark-favicon-changer/

• https://myhealthonline.sutterhealth.org/mho/favicon.ico

I am a Windows user with this same issue. I will supply a screen shot of what's on my end.

If the URL that is used to open a web page is different from the actual URL that appears on the location bar if the page has loaded then it is possible that you do not get a favicon.
You would either have to modify the bookmark URL to make it match the target URL or use the above mentioned Bookmark Favicon Changer extension to set a specific favicon to the bookmark.