How to append the text to the TextArea Widget?

You could try creating a custom service that would take the input as the current content of your textarea widget, then add another string to it.

Use the expression widget, this way you will reduce network traffic. It should have to inputs, one from the text area text attribute (call it input1) and one for the text you want to append (input2). The output should be a string and the exprerssion should be Output = input1 + input2, which should be bound to the text area text attribute. Evaluate the expression on the clicked event.