You don't need jquery

change-month-dropdown.jquery.js

// With Jquery.
$('.year-select').once('current-year').on('change', function () {
        var month = $('.month-select');
        var currentYear = (new Date()).getFullYear().toString();
        var selectedYear = jQuery('.year-select option:selected').val();

        if (currentYear === selectedYear) {
          var currentMonth = (new Date()).getMonth();

          month.find('option').each(function () {
            var optionMonth = parseInt($(this).val());

            if (optionMonth <= currentMonth) {
              if ($(this).is(':selected')) {
                $(this).next().prop('selected', true);
              }
              $(this).prop('disabled', true);
            }
          });

        }
        else {
          month.find('option').prop('disabled', false);
        }

        // Let other JS libraries know options are changed.
        month.trigger('change');
      });

      $('.year-select').trigger('change');

change-month-dropdown.vanilla.js

// With vanilla javascript.
var yearElement = document.querySelector(".year-select");

yearElement.addEventListener('change', function (event) {
  var monthElement = document.querySelector('.month-select');
  var currentYear = (new Date()).getFullYear().toString();
  var selectedYear = event.target.value;
  
  if (currentYear === selectedYear) {
    var currentMonth = (new Date()).getMonth();

    [].forEach.call(monthElement.options, function (item) {
      var optionMonth = parseInt(item.value);
      if (optionMonth <= currentMonth) {
        if (item.selected) {
          item.nextSibling.selected = true;
        }
        item.disabled = true;
      }
    });
  }
  else {
    [].forEach.call(monthElement.options, (item) => item.disabled = false);
  }
});

yearElement.dispatchEvent(new Event("change"));