Update, 8-21-17: the fix below (switching back to 32-bit Firefox) works great until you restart Firefox, then tada, it never works again. So I guess there is no answer short of trying every version of the LastPass add-on - really not a good idea when latest versions are patched for security vulnerabilities and so on and oh, LastPass, how completely unusable you are, let me count the ways.
Since discovering my fix only works until restart I've disabled LastPass and installed KeePass/KeeFox because I had online work to do and wasn't about to keep playing games with a broken password manager. KeePass has its own issues (mainly, when it stores more than one login for a site it tends to autofill the wrong one, leading to a lot of "copy username/copy password" clicking and pasting) but though it's not for the faint of heart (it's sort of an old-school program with about a gajillion options I haven't even glanced at yet) it does seem less batshit fucking insane to deal with, overall.
So, the LastPass blank dropdown menu and blank search results panel is very annoying. The dev hasn't updated the add-on since June and is responding to exactly zero complaints about this and other issues on his Firefox review page, though there might easily be dozens.
Which came to bite me, too, when Firefox finally let me have their latest multiprocess (e10s), 64-bit compatible version earlier this week (e10s is still automatically disabled if you install any add-on that isn't yet e10s capable); ever since I've had both LastPass problems, and saw others are having them, too [Example 1, Example 2, Example 3].
To fix these issues, just switch back to Firefox 32-bit. It's not even necessary to remove Fx 64-bit. It's actually better if you don't, so Firefox can just poke around in your profile folder and recreate the Firefox you've got in the 32-bit version you're about to get (just be sure to create a shortcut or a target that you can easily tell apart from the 64-bit icon).
32-bit Firefox runs LastPass perfectly, fixes the blank dropdown list of log-ins for each site and fixes search result panels showing up blank.
For everyone leaving bitter reviews [Example 1, Example 2, Example 3] and sharing the version number that allegedly works better [Version 4.1.62a]: I tried it in 64-bit Firefox, but it gave me all the same blank dropdowns as before.
My guess is the problems are not confined to any particular version. After I installed the May 31st version and saw the same issues it became clear the latest version is not at fault - it's 64-bit Firefox - and I'll gander that's no matter which version of LastPass going back to the earliest 56*-capable version you pick.
So if you've got 64-bit Firefox, try going back to 32-bit (here are the 32-bit installers. If you have automatic updates turned off, keep checking the directory for the latest). Run Firefox 32-bit with whatever version of LastPass you have and see if that fixes the problems.