On Windows pressing the space bar initiates a 'click' action on a button that has the focus. This is not working for a ToggleButton that is assigned to a ToggleGroup (without the ToggleGroup it does work). Here is a small test program to illustrate (you can comment out the ToggleGroup section to see the space bar working on the same buttons):

import javafx.application.Application;
import javafx.scene.Node;
import javafx.scene.Scene;
import javafx.scene.control.ToggleButton;
import javafx.scene.control.ToggleGroup;
import javafx.scene.layout.FlowPane;
import javafx.scene.layout.StackPane;
import javafx.stage.Stage;

public class ToggleButtonTest extends Application {

    public static void main(String[] args) {
        launch(args);
    }
 
    @Override
    public void start(Stage stage) {
     StackPane pane = new StackPane();
     pane.getChildren().add(createContent());
     stage.setScene(new Scene(pane));
        stage.show();
}
    
    private Node createContent() {
     FlowPane fp = new FlowPane(2, 2);
     ToggleButton tb1 = new ToggleButton("button 1");
     ToggleButton tb2 = new ToggleButton("button 2");
     ToggleButton tb3 = new ToggleButton("button 3");
     ToggleButton tb4 = new ToggleButton("button 4");
        
     // Comment out the following lines...
     ToggleGroup tg = new ToggleGroup();
     tg.getToggles().add(tb1);
     tg.getToggles().add(tb2);
     tg.getToggles().add(tb3);
     tg.getToggles().add(tb4);
     // ...to here, and the space-bar works again to select the buttons
    
     fp.getChildren().add(tb1);
     fp.getChildren().add(tb2);
     fp.getChildren().add(tb3);
     fp.getChildren().add(tb4);
     return fp;
    }
}