I'm not sure if you have but please, read this first: www.cregox.com/your-home
I don't think squarespace ever had or will have an API. Or any way to get data in (the server side). We can only work with content and client side scripts. Even the "developer" platform is only meant to develop templates (all client side).
That being said, it looks like shipstation, for instance, does offer squarespace integration. http://www.shipstation.com/ It's not impossible to think how that works, they have their own servers and you need an account there. The only thing that will never happen this way is an integration between their commerce and squarespace's, for stock control and all that stuff.
I could work something out for you, but I can't give a good estimate on final quote. What I can do is tell you to get something working (using something like shipstation) which I myself would be glad to work with, it will be less than U$432. Then we can move from there if you need more.