I require AES256 encryption/decryption in a commercial web application. Currently everything is good with a key size of 128. This is not satisfactory cryptographically so my
You are either stuck with the weak encryption or a potentially problematic step for users if you stick to the SunJCE.
There is obviously no problem importing an AES library, there is just a problem using it using an instance of Cipher. If you have a specific piece of software that does not depend on JCA, you can for instance rewrite it to use the lightweight crypto API of Bouncy Castle.
Note that many other parts of the Bouncy API themselves depend on the JCE. The lightweight API is also trickier to use and less documented/tested than the SunJCE.
The Bouncy Castle lightweight API is pretty large as well. It contains a lot of functionality that you won't need. So it is probably too large for your applet. If it is I would advice you to create a new library that only contains the specific classes that you need from Bouncy Castle. The Bouncy Castle is fortunately very liberally licensed. As long as you keep the copyright statements etc. in place, you can easily split it off.