Pure Java solutions / libraries for converting videos to different formats might be a bit hard to get by. There are a lot of open source / free converters out there [like ffmpeg] which you can use in conjunction with your application to achieve the desired task since re-writing the entire thing in Java would be a significant act.

One of the ways to do it:
• Use your web application as a front end for allowing users to upload their videos.
• When storing the videos, either store them directly to your File system or use a Database; depends on how your project is engineered.
• Run a background process which would pick up the file from a specified directory, convert it and put it in another directory [or table if using Database].
• The background process would run a command line version of the file conversion utility for converting files.

Another way would be to create a thin wrapper around the open source libraries out there which are written in C or C++ and make native calls using JNI.

But on a final note, this is by no means a simple project and having an architect on your team who *knows* things really helps.