As you may already know, Android is moving away from Flash.
None of my Android devices today give me access to the File Manager in Webmin.
I hope, someday, there is a solution for Android users in this area.
Thank you for a great product! 
The File Manager does not use Flash. It uses Java (almost as bad!).
We do have plans/hopes to implement a JavaScript file manager, but our in-house JavaScript skills aren’t the strongest, and we don’t have a budget to hire someone. We’re kinda hoping a really good Open Source JavaScript file manager frontend shows up, and we can extend it and integrate it into Webmin. 
What about AjaxPlorer? It’s very good, don’t know though if it could be integrated.
Thank you for all your replies, and thank you for the clarification about it requiring a Java plugin (and not a Flash plugin.)
When I am doing something with my servers, I am usually on one of my PCs and use ssh. It was just this one time when I wanted to quickly check if a certain file was created and I was nowhere near a PC to do that. I thought I’d just use my handy Android tablet to login to Webmin and look through the “File Manager” when I discovered the “Missing Plugin” error on the page.
In my frustration, I decided to come here and report it… but now I can appreciate that it is potentially a LOT of work for very little usefulness.
If anybody here finds herself/himself in a similar situation, and has to rely on an Android tablet or phone in an emergency, just search for a suitable ssh client app for your device and don’t bother about the File Manager on Webmin, at least not with your Android devices.
If you haven’t heard of it yet, check out ConnectBot for Android. When I finally figured out how to use it, I was grateful I had this problem in the first place. Now I have a ssh client in my hand all the time!
Cool! Yeah I use ConnectBot as well for administration on the go.
Also, if you’re looking to just browse around the filesystem a bit on your Android device, another alternative is to use an SFTP client.
I haven’t used this one before, but something like Android’s TurboClient (available in Google Play) should do the trick there.
-Eric