The second implementation option for the article: https://kolmogorov.pro/integration_google_recaptcha_v3_in_site_on_vue_js

recaptcha.js

let reCaptcha = new Promise((resolve, reject) => {
   const $script = document.createElement('script')
   $script.src = 'https://www.google.com/recaptcha/api.js?render=ВАШ_КЛЮЧ'
   resolve(document.head.appendChild($script));
 
   setTimeout(() => reject(new Error("Google reCaptcha не инициализирована")), 3000);
});
 
reCaptcha
.then(
   result => {
       setTimeout(() => {
           grecaptcha.ready(function() {
               grecaptcha.execute('ВАШ_КЛЮЧ', {action: 'homepage'}).then(function(token) {
                   axios
                   .post('/functions/recaptcha.php', {
                       token: token
                   })
                   .catch(() => {
                       console.log('POST-запрос в API Google не был отправлен.');
                   })
 
               });
           });
       }, 1000)
   },
error => {
   console.log(error);
});