This is a regression of the issue previously described in another post here. The symptoms are identical: when clicking the extension's icon on the toolbar a blank rectangle is displayed instead of the main menu. In regular, non Private mode, the menu is rendered correctly.

The versions are:
Firefox - 100.0
LastPass - 4.95.0.3

Just like in the original post, uninstalling/reinstalling the extension doesn't solve the problem. Nor does a previously known workaround of right-clicking the popup and selecting This Frame -> Open Frame in New Tab, which used to do the trick.

However, if you do open the frame in new tab/window, right-click and pick Inspect, you'll see that there's actually valid HTML provided for the frame:

Fragment of the LastPass menu's HTML code

There, there's the line <div id="container" class="">...</div>, by clicking on which you can see the CSS properties of the div element:

CSS for the container

Ironically, it's marked with visibility: hidden, which seems to be the root cause of the issue, since disabling this property actually makes the whole LastPass menu render just fine. Not convenient enough as a workaround, but still nice to have as a last ditch effort.

Anyways, if there are LastPass devs in this subreddit, hopefully you can fix this asap, as not being able to use LastPass in Private mode is just a shame.