Contact Form 7 has some Datepicker support via the new HTML5 date input type (see https://contactform7.com/date-field/) but this is currently only supported on some browsers – see https://www.wufoo.com/html5/types/4-date.html
Contact Form 7 does provide jQuery UI-based fallback for the date and number input fields. By using this solution, you can provide calendar UI for the date field and spinbox UI for the number field, respectively, even with Firefox or Internet Explorer. By default, this fallback feature is disabled because it loads extra JavaScript and CSS (makes for poor performance) and it is only necessary for websites that use the date or the number input fields. If you use the date or number input fields and wish to use this jQuery UI-based fallback feature, add the following code into your theme’s functions.php file and activate the feature:
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add_filter( 'wpcf7_support_html5_fallback', '__return_true' );
See https://contactform7.com/faq/does-contact-form-7-support-html5-input-types/
