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No. From reading the first topic on this it is not possible because to run runescape, you need to download some sort of software to your Wii and the Wii cant run what Runescape and Java needs to download.
It is fully possible if you Mod java and save it to your SD card. However, this could have strange effects.
I would also like to mention typing would be availble, providing that you use a USB based keyboard (like the one I'm using on the wii right now). They are easy to find on apple.com