There are multiple problems I can see:

1. As VLAZ states in the comment, you're trying to get the .data.id on a value that is already an id
2. Creating a sequence of promises should be done using the tools provided by the Promise util (https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Promise/all might be what you are looking for).

For instance I would go with something along those lines (not tested):

const contacts = document.getElementsByName('newContact');
const contactName = document.getElementsByName('contactName');
const contactTitle = document.getElementsByName('contactTitle');
const contactPhone = document.getElementsByName('contactPhone');
const contactEmail = document.getElementsByName('contactEmail');
let contactIDArray = new Array();

const requests = []

for (let i = 0; i < contacts.length; i++) {
  const name = contactName[i].value;
  const title = contactTitle[i].value;
  const phone = contactPhone[i].value;
  const email = contactEmail[i].value;
  var id;
  promise = createContact(name, title, phone, email).then(function (res) {
    const id = res.data.id;
    alert(res.data.id);
    return id;
  });
  requests.push(promise)
} 

Promise.all(requests).then(ids => {
    // This should contain the array of ids
    alert(ids)
})

Edit: Also, be careful about sharing result between synchronous and asynchronous calls, usually you would stay in the async world once you started your async tasks, as stated here: How to return data from promise

I'm trying to return the ID of the created object into an array so that array can later be inserted into another object

Based on your request, I have a few things for you:

1. Like jkoestinger's answer, do not use async/await or asynchronous functions (.then()) inside a loop unless you really need it.
2. Do not return a property directly from the async call, you should return the whole object and get the ID. This is a good practice.

Please see the modified code below:

index.js

let contactPromises = [];

for (let i = 0; i < contacts.length; i++) {
  const name = contactName[i].value;
  const title = contactTitle[i].value;
  const phone = contactPhone[i].value;
  const email = contactEmail[i].value;
  
  contactPromises.push(createContact(name, title, phone, email));
}

let contacts = (contactPromises.length > 0) ? await Promise.all(contactPromises) : [];
let contactIds = contacts.map(contact => contact.id);
// use your contactIds here

axios

let res = await axios({
  method: 'POST',
  url: '/api/v1/contacts',
  data: {
    contactName,
    contactTitle,
    contactPhone,
    contactEmail,
  },
});

return res.data;   // <--- return res.data only, not the res.data.id