Here's a list of 22 Best Shopify Fulfillment apps, check them out and find the most suitable app for your need.
Hope it'll help
You can auto fulfill orders with Order Automator. You create fulfillment rules based on the order contents and it will handle it automatically.
I led the development on the app, so if you need any custom features beyond what the standard app offers, just reach out through the app support and ask for Joe, I'll take care of you.
It sounds like our app, EZ Fulfill, could help with your situation. EZ Fulfill allows you to manually or automatically fulfill your orders with tracking numbers. You also have the option to send email notifications to your customer with their tracking details.
On the automation side, we support FTP, SFTP and Dropbox at this time and you can fulfill your orders as frequently as every hour. We also support fulfilling orders by SKU or Order. When fulfilling orders automatically we allow you to provide a set of variables to ensure we can read files even if they are provided with the current date, you can read more about how this works here. Our app will also alert you if there are any errors fulfilling your orders so you can set the automation up and just let it run. If you have an further questions please don't hesitate to let us know.
Hopefully you'll find the right solution!