Your mobileMenu variable is working, the functionality not working is UI changes depend on mobileMenu variable. Modify your code to this:

const myDropdownMenu = document.getElementById("dropdownMenu");
const openNav = document.getElementById("openNav");
const closeNav = document.getElementById("closeNav");

let mobileMenu = false;

function toggleMenu() {
  mobileMenu = !mobileMenu;
  if (mobileMenu) {
    myDropdownMenu.style.display = "flex";
    closeNav.style.display = "flex";
    openNav.style.display = "none";
  } else {
    myDropdownMenu.style.display = "none";
    openNav.style.display = "flex";
    closeNav.style.display = "none";
  }
}

/*
if (mobileMenu) {
  myDropdownMenu.style.display = "flex";
  closeNav.style.display = "flex";
  openNav.style.display = "none";
} else {
  myDropdownMenu.style.display = "none";
  openNav.style.display = "flex";
  closeNav.style.display = "none";
}
*/

The if/else should be within the function, like this:

function toggleMenu() {
    mobileMenu = !mobileMenu;

    if (mobileMenu) {
        myDropdownMenu.style.display = "flex";
        closeNav.style.display = "flex";
        openNav.style.display = "none";
    } else {
        myDropdownMenu.style.display = "none";
        openNav.style.display = "flex";
        closeNav.style.display = "none";
    }
}

This is because otherwise the if/else only runs once (on load), whereas we want it to run every time the function is triggered.

Your if/else is in the wrong place so it doesn't execute when the function is executed. In order for the if/else to execute when the function is executed, it has to be in the curly brackets of the toggleMenu function (inside the function).

The code below fixed you problem by moving the if/else statement into the right place (into the function) so it executes when the function is called.

const myDropdownMenu = document.getElementById("dropdownMenu");
const openNav = document.getElementById("openNav");
const closeNav = document.getElementById("closeNav");

let mobileMenu = false;

function toggleMenu() {
  mobileMenu = !mobileMenu;

  if (mobileMenu) {
    myDropdownMenu.style.display = "flex";
    closeNav.style.display = "flex";
    openNav.style.display = "none";
  } else {
    myDropdownMenu.style.display = "none";
    openNav.style.display = "flex";
    closeNav.style.display = "none";
  }
}

      <div>
        <a href="index.html"> <h1>CAVVD</h1> </a>
        <div id="mobileNav" class="mobileNavigation">
          <button id="openNav" onclick="toggleMenu();">
            Open
          </button>
          <button id="closeNav" onclick="toggleMenu();">
            Close
          </button>
          <div id="dropdownMenu">
            <a href="index.html#sobreNosotros">Sobre Nosotros</a>
            <a href="index.html#reuniones">Reuniones</a>
            <a href="ForodeConsultas.html">Foro de Consultas</a>
            <a href="index.html#HaceteVoluntario">Hacete Voluntario</a>
            <a href="Fotos.html">Fotos</a>
            <a href="index.html#contacto">Contacto</a>
            <a class="Dona" a href="index.html#Dona!"> Doná!</a>
          </div>
        </div>
      </div>