Sort a ListBox on a Dialog

var nLocale = Context.getSelectedLanguage(); //get the current locale
var selModelList = ArisData.getSelectedModels(); //get the list of selected models

var modNames = []; //an array for model names as strings

selModelList.forEach(function(value){ //for each of the selected models
    modNames.push(value.Name(nLocale)); //put the name of the individual model in the list of model names as strings
});

function myDialog(){ //function for a new dialog
    
    this.getPages = function(){ //describes how the dialog is supposed to be built
        var userdialog = Dialogs.createNewDialogTemplate(0, 0, 680, 320, "Output sequence"); //stole this from your question...
        
        userdialog.Text(10, 10, 460, 15, "Select the output sequence.");
        
        userdialog.Text(10, 25, 460, 15, "Click on \'OK\' to finish the selection.");
        
        userdialog.OptionGroup("Options");
        
        userdialog.OptionButton(20, 55, 360, 15, "Sort alphabetically", "Opt1");
        
        userdialog.OptionButton(20, 70, 360, 15, "Sort manually", "Opt2");
        
        userdialog.ListBox(20, 90, 586, 220, modNames, "list");
        
        userdialog.PushButton(610, 90, 25, 25, "<SYMBOL_ARROWUP>", "MoveUp");
        
        userdialog.PushButton(610, 280, 25, 25, "<SYMBOL_ARROWDN>", "MoveDwn");
        
        userdialog.OKButton();
        
        userdialog.CancelButton(); //till here
        
        return [userdialog]; //return the single dialog template template
        
    }
    
    
    this.MoveUp_pressed = function(){ //the function that describes what the MoveUp button does
        var listBoxObject = this.dialog.getDialogElement("list"); //get the dialog element list (it's needed multiple times, don't want to call this.dialog.getDialogElement("list") again and again
        
        var futureOrder = listBoxObject.getItems(); //get the array of strings that currently populate the ListBox
        var currSelectionIndex = listBoxObject.getSelectedIndex(); //get the index of the currently selected element in the ListBox
        
        if(currSelectionIndex != 0){ //if the current selected element is not the topmost (wouldn't be able to move that)
            this.dialog.getDialogElement("Options").setSelection(1); //set the correct OptionButton (radio Button). I think this is just convenient.
            
            var tempSave = futureOrder[currSelectionIndex]; //save the string value of the current selection to a temp variable
            
            futureOrder[currSelectionIndex] = futureOrder[(currSelectionIndex-1)]; //make the text in the current selection the text that stood originally above the selected element
            futureOrder[(currSelectionIndex-1)] = tempSave; //make the text above the selection display the text of the original selection
            
            listBoxObject.setItems(futureOrder); //the array element order was changed sucessfully, now put it back into the dialog
            listBoxObject.setSelection((currSelectionIndex-1)) //move the selection one element up, allowing the user to click the button again and again to move the same element upwards
        }
        
        

    }
    
    this.getResult = function(){ //function that describes what should be returned
        return this.dialog.getDialogElement("list").getItems(); //return the current list of sting elements of the ListBox
    }
    
    
    
}

var result = Dialogs.showDialog(new myDialog(), Constants.DIALOG_TYPE_ACTION, "Output sequence"); //show the previously described dialog as a single-page-dialog

Code is  commented, should describe the functionality.

I brought your dialog into the shape described in the ARIS script help (bundle all the methods of the dialog into a single method and use a single Dialogs.showDialog(...) method to display the dialog).

Edit: corrected small mistake in the code comments