Ted here from Shopify, thanks for reaching out to us!
When a merchant closes a store, we would normally recommend removing all apps from the account before closing. We suggest this to avoid any additional charges that might occur when you re-open the store or in case the apps continue to charge an account after it has been closed, as mentioned in this guide. That said, it looks like the app was not removed before closing and it started a new billing cycle when you re-opened the account.
Shopify are unable to provide a refund without the app developers permission. What I'll do here is I will help contact the app developers via email on your behalf and ask them for a refund. Please know that a refund is not guaranteed but I will also CC you in the email so you will receive updates as soon as possible. Please make sure to remove the app in question from your admin to be considered for a refund.
If there is anything else I can help you with, please let me know, I'm happy to help!