I work in this space and have some ideas to add:
The incoming order fulfils automatically and gets sent to the supplier accordingly
Order Automator app will do that - you can create rules to fulfill line items, also to send notifications if you want. So if a product comes in for Vendor A, then you can have it autofulfilled and send an email to that vendor.
My team built that so we can also customize if you need it (for example some of the points you mentioned below).
Inventory gets updated at the supplier and at my shop.
This will happen automatically on your shop. For “at your supplier”, it will depend what system they are using. If they just need a notification, then nothing additional needed. If they’re using software, the first step would be to see if their software allows API connection. If yes, you / we could write some code to update the inventory count on new orders within an app.
sending emails towards customer would be a plus
Shopify handles this automatically.
sending the invoice to the customer would be a plus
One idea, you could include the invoice in the order confirmation email. Or have an app send them an invoice (might be restricted to certain formats, for example, a csv would be easy but a PDF may be more difficult to generate from code).
any integration into accounting software would be a plus
Depends what the accounting software is. If they have an API then yes you can write code in an app to communicate with it when new orders come in.
Summary, anything’s doable, an existing app is your best solution but if you can’t find what you’re looking for and need a 100% custom app feel free to reach out, my team builds both public and private apps, we solve problems like this all the time.
Hope that helps.