Hyde here from Shopify. Great question!
It is usually possible with a Shopify free theme to indicate what thumbnail you want to be displayed whenever you share your store link (more here
However, if you're using a third-party theme you may need to reach out to the theme developers directly about this. You can usually find their contact information by going to Online Store > Themes > Customize > Theme Settings and scrolling to the bottom.
I hope that helps! Let me know if you have any other questions.
All the best, Hyde.