Hi,
Your requirement is pretty clear and straightforward to implement. When an order is placed on ecwid it'll send a push message to the bridge application. Once the message is received it'll package that message to make it suitable for linnworks and create the order there, too. Same flow in that or reverse order might be applied for other actions such as shipping, ect.
Before I can start I expect you to provide accounts for both that I can use for development. Once provided, I can deliver the project within 5 days at most. I'm planning to use php to implement the integration which you'll need a server to run, a $5/month DigitalOcean server would be more than enough. Also, I'm not sure how complex that's supposed to be but depending on that my bid and deadline might change.
Thanks
Ibrahim