Platform for API shopping cart integration

I work for a 3PL (3rd Part Logistics company), warehousing and shipping goods for 100's of clients.
Most warehouse orders are received electronically via EDI, XML, Flatfile... and mapped into our WMS (Warehouse Management software) through a middware mapping platform.

I need to get current and offer an API solution. I need to be able to quickly and easily talk to shopping cart API's (Shopify, Magneto, WooCommerce), as well as offer our own API that clients can interact with.

I don't have the tech skills in house to design or build this. Is anyone aware of of an existing software platform I could purchase that would support both types of API?
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JDCamAsked:
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Scott Fell, EE MVEDeveloper & EE ModeratorCommented:
It looks like Shopify, Magento and WooCommerce all have 3rd party providers for EDI but I would think that is not the type of integration you are looking for. If you want to have a cart sync with your back end, that probably means you need to know if something is available in stock prior to purchase and where to ship if it is.  You may need to build your own bridge application.
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