Currently, the InputBoxText property of the RadChat control has a read-only purpose. It is updated only when you update the text via the UI. However, the property has also a public setter which implies that it can be set, but this doesn't work. 
Allow setting the InputBoxText property, which should update the text in the input TextBox.

For the moment, you can get the RadWatermarkTextBox element using the FindChildByType<T> method and set its Text property manually.

var textBox = this.chat.FindChildByType<RadWatermarkTextBox>();
textBox.SetCurrentValue(RadWatermarkTextBox.TextProperty, "new text")