Weebly does not have any direct integrations with shipping fulfillment companies that are in-place at this time. However, there are companies that have used ShipStation with Weebly. Weebly also now integrates with Square, which integrates with Stitch Labs and ShipStation. Those are probably your best bets. Other shipping fulfillment services often allow you to configure a custom API, which will basically create a custom solution that helps you integrate the two services. Check with your preferred provider to see what they can do for you.
If you have any other questions, let me know!