First, understand that Lightroom does not store images in its catalog, but instead simply records a location pointer. So organizing within Lightroom is very powerful, if you take advantage of the power. But you have to let go of thinking about organization equalling folders on a hard drive.
To answer your question: No Lightroom does not allow you to organize Folders when you are not connected to the external hard drive. The Folders view within the Library Module effectively mirrors what is on the hard drive and you can physically move files within the Library>Folders section. But Lightroom doesn't do a scan of those folders, so if you want to move files outside of Lightroom, you must tell Lightroom to check the folders and update the location pointers. You can do this by right clicking on the folder and selecting 'Synchronize Folder'. However this only updates Lightroom if you do a move or reorganize outside of Lightroom. It doesn't cover your case of reorganizing within Lightroom and having this physically take effect once you are connected to your external drive. Since Lightroom is simply using the underlying OS to move files, the fact that your external drive is NOT connected means that it is impossible to reorganize within Folders without your external drive connected.
But, organization is where the power of Lightroom comes in. You see, Lightroom does not care where your photos are located. It shows you the folders as a convenience for you. But, it isn't actually accessing any of the files in those folders...it is simply mirroring what those folders looked like, last time it checked.
Instead of worrying about what is in physical folders, leverage the Collections view within the Library module. Think of it as 'virtual folders'. With Collections, you can organize to your hearts content, without being connected to the external drive! But you have to give up the notion that organization means very specific folder names on a hard drive. Sure, keep some order, such as folder names as dates, but if you want to use event names or client names, then leverage Collections instead.
So, close the Folder view in Navigator, and embrace Collections and Smart Collections.