Okay, that advice above may be useless since it looks like apps installed via APK files may not have a "move to SD card" option. At least it was disabled for me when I tried it out on the BlueStacks Android emulator, even though it's not disabled in general.
However, I did notice that the Lemmings files (eg. levelXXX.dat, actually I think it needs all the DAT files from the game) aren't included in the APK anyway, so you'd have to explicitly put them in the location the program expects to find them. So more likely it could be a problem due to the exact way you got those files onto your device. Your error specifically is a permission-denied problem. I got it working fine for me after I put the files in the location mentioned in the "file not found" (as opposed to your "permission denied") error message I first saw when the files aren't there yet.
When I tried it out, because it was on an emulator and I don't know how to put files directly onto the emulated SD card outside of Android, I actually did that through Android: I email the files to myself as a zip file, download the zip file from the web browser on Android, then use a File Manager app on Android to extract the files to the location desired by the game (the location it gave in the error message when the files aren't there yet). A more natural way if you are using a real device is to just pre-populate the SD card with the right files and folders on your PC, and then put the card back onto the device, and since IIRC SD card uses FAT32 file system which shouldn't have permissions, it should work just as well as the more cumbersome email method I went through. Nevertheless, I gave you two methods you can try to get working. [Sidenote: if you are using a real device, I think you can also just plug the Android device to your PC via USB, and there should be options for you to make your Android device appear as USB storage to the PC, which I think (?) would expose the SD card as well, and then you can copy files and folders onto the SD card that way without having to take it out of the device and putting it back in.]
Oh and if your device or emulator doesn't actually have an SD card at all when you hit your error, please retry with an SD card (real or emulated). I bring this up because again, the file path in your error message doesn't look like an SD card file path, even though in my case and in Johanne's case it was. The only explanation I can think of is if on your device/emulator there is no SD card at all, which ends up causing Droidlings to look for the files in internal storage instead, which may have stricter access permissions. If this is the root cause then once you insert the SD card (and potentially kill/restart the app or maybe just reboot your device), I expect the error message should change to "file not found" instead of "permission denied", and should show a different file path, likely one that contains "sdcard" somewhere in its text.
