Javascript If / Else Loop; асинхронная проблема

Если я правильно вас понял, вы хотите сделать следующее:

var ref = firebase.database().ref();

$("#connectNextUser").click(function() {
  // query how many customers are waiting
  ref.child("customerWaiting").on("value", function(snapshot) {
    // as soon as you have the result  then get the numChildren
    var customerWaiting = snapshot.numChildren();
    console.info("There are " + snapshot.numChildren() + " customers waiting");

    if (customerWaiting > 0) {
      // show the modal if customerWaiting > 0
      $("#myModal").modal();
      console.info("connect");
    } else {
      console.info("There are " + customerWaiting + " employees waiting");
      console.info("no connect");
    }
  });
});

Если вы хотите использовать await / async, тогда ref.child("customerWaiting").on("value", resolve) должен поддерживать обещания, или вам нужно преобразовать его в один:

var ref = firebase.database().ref();

$("#connectNextUser").click(async function() {

  var snapshot = await new Promise((resolve, reject) => {
    ref.child("customerWaiting").on("value", resolve)
    // you should also handle the error/reject case here.
  })

  var customerWaiting = snapshot.numChildren();
  console.info("There are " + snapshot.numChildren() + " customers waiting");

  if (customerWaiting > 0) {
    $("#myModal").modal();
    console.info("connect");
  } else {
    console.info("There are " + customerWaiting + " employees waiting");
    console.info("no connect");
  }
});