My name is Alex, I'm a guru here at Shopify!
Understood! Most apps that you'd find in the app store would handle syncing with either a larger marketplace that we integrate with, or with another Shopify store - successfully syncing with another store that isn't on the Shopify system is a somewhat more complicated matter, as every service will have its own system for dealing with its stock.
That said, I have found one app that may help you out - it's called Stock Sync - Simple inventory update, and it allows for syncing your products using a variety of methods. This is a third party, paid app, so we're not the experts in its function ourselves - I'd suggest reaching out to the developers directly on that app page to make sure that this solution will work for your needs!
Hope that's helped! Please feel free to let me know, and you can always give us a call or start a live chat at any time. We're open 24/7 for your convenience and always happy to assist!
Alex | Shopify Guru