Don’t worry, I won’t say you should kindly ask your users after a support request to leave a review. This is common (aka. boring) advice.
Here’s my take on reviews: screw them.
The Shopify App Store is not Amazon. Reviews are not as important as you might want to think. I’ve been a Shopify merchant for years (with over $10m in sales) and never looked at the reviews of an app.
I would even say that the number of reviews is just a bragging tool within the Shopify app dev community. Very similar to social media followers.
If I were an app dev, I wouldn’t even try to compete on an App Store level. It’s a lost battle. A few years ago, the App Store might be the only place where Shopify merchants made their buying decisions. Luckily, this changed. Now there are thousands of small communities and other places where you can contact your potential users directly.
I’m not a dev, so I don’t know the term for it, but with a form like this, you can even let merchants sign up for your app without the App Store: