subscribe-able calendar plugin?

I create websites and have a client who I set up WordPress as a CMS for their site. They are a church and need a calendar that their church members can subscribe to from their website. To me, this seems like a simple request, but nothing I have found has been just what we want. Ideally, they would enter their email somewhere and every time there is an event, it would send them an email reminder. Additionally, if there is a new event added or taken away, they would also get an email. I plugged in “The Events Calendar” plugin, which allows for ical import, but it looks like the user would manually have to visit the website and continually “import” it to get the newest additions.

If anyone has ANY suggestions, I am ALL ears. I am pretty desperate at this point. Oh, and it actually doesn’t have to be a WordPress plugin. If you know of a calendar that does this that is unrelated to WordPress, I would love to hear about it, too.

Thanks for any help!