Error: No reCAPTCHA clients exist (reCAPTCHA v3)

Have you tried loading the script before trying to send the request?

<script src="https://www.google.com/recaptcha/api.js?onload=ReCaptchaCallbackV3&render=site_key"></script>
<script type="text/javascript">
    var ReCaptchaCallbackV3 = function() {
        grecaptcha.ready(function() {
            grecaptcha.execute("site_key").then(function(token) {
                console.log("v3 Token: " + token);
            });
        });
    };
</script>

I believe this error occurs when the reCaptcha api.js loads, but your container is not present on the page yet (at least for v2). I had this error occur in a React app when I navigated to the page rather than loading it as the first on. Instead of using render=explicit and using a global namespace onLoadCallback, I was able to resolve it by rendering the captcha element manually.

Instead of creating a <div class="g-recaptcha"></div>, give the container div an id only (<div id="recaptcha-container"></div>) and render it in your JS code (e.g. in componentDidMount for a React app):

grecaptcha.ready(function() {
  grecaptcha.render("recaptcha-container", {
    "sitekey": "your-site-key"
  });
});