Sendy is open source (albeit not free) and running on your own server. Ben's response says because you are using a customized webform instead of sendy's provided forms, that you need to handle the captcha code on the form and siteverify on the backend on your own.
Before sendy had recaptcha support you would have to modify the subscribe code to include a call to captcha siteverify. Here's some PHP code for how you call siteverify before proceeding with something: https://www.adam-bray.com/2018/04/02/adding-recaptcha-with-p...
You can avoid running your own backend altogether now anyway since SES supports bulk template emails. You can store all your form submissions using a lambda/cloudfunction/webworker to verify the captcha and store the list. When you want to send email you can pull your list into something running on your laptop and then invoke the SES bulk templated email from there. You can use lambdas for pixels and custom links that update records for those users. I wrote my own angular-firebase version of this