I am going to answer my question because I found a solution.
First of all I added an event listener for all input tags on webpage, after page loaded:
private void addEventListenersToDOM() {
webview.getEngine().getLoadWorker().stateProperty().addListener(new ChangeListener<State>() {
@Override
public void changed(ObservableValue<? extends State> ov, State oldState, State newState) {
if (newState == State.SUCCEEDED) {
JSObject win = (JSObject) webview.getEngine().executeScript("window");
win.setMember("javaFXVirtualKeyboard", new JavaFXVirtualKeyboard());
String script =
"var inputsList = document.getElementsByTagName('input');"
+ "for (var index = 0; index < inputsList.length; ++index) { "
+ "inputsList[index].addEventListener('focus', function() { javaFXVirtualKeyboard.show() }, false); "
+ "inputsList[index].addEventListener('focusout', function() { javaFXVirtualKeyboard.hide() }, false); "
+ "}";
webview.getEngine().executeScript(script);
}
}
});
}
And the key point, how I triggering the keyboard displaying and hiding:
public class JavaFXVirtualKeyboard {
public void show() {
FXVK.init(webview);
FXVK.attach(webview);
}
public void hide() {
FXVK.detach();
}
}
One note: FXVK class is not an API so we get a warning message in all cases but it works without any bug.
Discouraged access: The type 'FXVK' is not API (restriction on required library 'C:\Program Files\Java\jre1.8.0_91\lib\ext\jfxrt.jar')