Example : Using Alpine Js to set WordPress AJAX method.

Form_wordpressAjaxwithAlpinejs.html

<form method="POST" x-data="checkoutGuestForm()" @submit.prevent="submitData">
  <p>
  <span>
    <input type="email"
           placeholder="Enter Email"
           name="email"
           x-model="formData.email"
            required
    >
  </span>
  </p>
  <p x-show ="existingCustomer">
  <span class="passowrd">
    <input type="password"
           placeholder="Password"
           name="password-checkout"
           x-model="formData.password"
    >
  </span>
  </p>
  <p>
    <span><button :disabled="loading" x-text="buttonLabel"></button></span>
  </p>
  <p>
    <span class="error" x-text="message"></span>
  </p>
</form>

wordpressAjaxwithAlpinejs.js

// Need to use window otherwise, Alpine JS will not recognize this. Out of scope
window.checkoutGuestForm = function () {
  return {
    formData: {
      email: '',
      password: '',
    },
    message: '',
    submitData() {
      fetch(localizedObject.ajaxurl, {
        method: 'POST',
        headers: { 'Content-Type': 'application/x-www-form-urlencoded; charset=utf-8', 'Cache-Control': 'no-cache' },
        body: 'action=checkout_guest_form&nonce='+crush_ajax.nonce+'&email='+this.formData.email+'&password='+this.formData.password,
        credentials: 'same-origin',
      })
        .then((response) => {
          response.json().then(data => {
          // This is where you get your response back in "data" variable
          });
        })
        .catch(() => {
          this.message = 'Ooops! Something went wrong!'
        })
        .finally(() => {
		// Fires when you have finally got the response.
	  })
    },
  }
}

This code is not tested to run out of the box, there are certain data that I have not added. You will have to remove certain x-data variables in order to make it work.