You might want to look into Klaviyo. You can setup a number of flows there - very user-friendly. However, it is a paid subscription. What you're looking to do, is not going to be your typical free app functionality. https://www.klaviyo.com/ (Shopify recently partnered with Klaviyo in replacement of Mailchimp).
Shopify does make a really good Product Reviews app of which can be found here: https://apps.shopify.com/product-reviews
What this app does is send customers an email notification where they can rate and review the product they purchased directly from their email inbox. You also have the ability to monitor those submissions. Nothing has to be published to your site unless you want it to be. You can set these to automatically publish to the site but there's never recommended :)
"What this app does is send customers an email notification where they can rate and review the product they purchased directly from their email inbox. "
I didn't find this option..
You can check about its Review feature here: AiTrillion's Review Feature.
For more details & if you have any issues, you can contact at firstname.lastname@example.org