Ash here from the Shopify team. Welcome to the Shopify Community!
This definitely sounds like a problem. I'm happy to help get this sorted out. Usually when the "country" drop-down appears blank, it's due to not having any shipping rates set up in your store. The "country" drop-down in the checkout process will automatically populate with the countries that are set up in your Shipping settings. If you don't have any shipping rates set up, then the checkout doesn't have any countries to list as options.
With that being said, in order to identify what the issue is and determine our next best steps, I'll need to get some more information from you. Can you please reply to this post and let me know the following:
Once I have the above information, I'll be able to dig in deeper to this issue. Looking forward to hearing back and getting this resolved!