Recaptcha

The calendar feature is not accessible to screen readers either so I guess recaptcha is the least of the issues with this plugin. I will look for an alternative.

Hello.
Its most probably that you are having some JavaScript error at that page. Please recheck your page for any JavaScript errors (usually for see what JavaScript error you have you need to press Ctrl {+} Shift {+} J for showing browser console) and try to fix them.
You can check for the most common issues on the FAQ page https://wpbookingcalendar.com/faq/ at the troubleshooting section.
You can also try to deactivate one by one all your active plugins at the WordPress > Plugins menu. Then retest it again.
If its not help, please deactivate your active theme and active the default WordPress theme. And then retest it again. May be there exist some conflict with some active plugin or actual theme and just need to find the reason of that issue.
If you still will have this issue, please send the link to the page with booking form, where you are having an issue, I will check it then.

No it has nothing to do with Javascript. The calendar feature can not be focused on when navigating with a keyboard and the captcha has no label for screen readers to hear.