The WebBrowser control isn't going to give you direct access to methods on a Java applet, AFAIK (you might be able to fudge your way through because I can't imagine that there isn't ActiveX involved, but I don't think that's the best answer).
Rather, what I would do is have a fuction in Javascript that accesses the value of the textbox from the applet (assuming that the control in the applet exposes it publically). It would assign the value to a property on the extern object exposed by the window object in Javascript.
Then, you would create your own class which you would assign to the ObjectForScripting property on the WebBrowser control which exposes that property.
Finally, when you click your button, you would call the script (you can assign the function to your same object, if you want) which will assign the value, then get the value from the instance that you set to the ObjectForScripting property on the WebBrowser control.
There are other ways you can do this as well, you could have the Javascript method write the text to a hidden element, then browse the DOM model to get the value, etc, etc.
In the end, whatever solution you have will more than likely involve a combination of DOM traversal and setting the ObjectForScripting property, as well as Javascript in the HTML hosting the applet.