Could you help me please with my issue?
We have our own Shopify store and want to import inventory data from our Partner's Shopify. If the products on the Partner's site are changed, we want the changes to occur on our site as well. If the product is deleted from the Partner's website, then it should be deleted from our store as well.
From what should we start? Maybe API or some App?
There are different apps available on app store to sync between stores however usually you need to be owner of both.
We have our own product data management app that lets Shopify stores share their products with other users and can select what data flows through. On the other end you decide how to treat the data.
For deleted items you probably just change status to draft but would depend on the other supplier how they manage their products.
In your case the partner Shopify store would need to set this up and share data form their side with you so a bit more complicated than just installing an app in your store.
You could test our product information management app but the store that is sharing the account would need to set things up and then share their data with you so you can then automate it into your store.