I have a set of ZBAPI structures on a server of a company in which we developed and installed a custom application.
I am having to use a lot of the same BAPIs in the same application for a different company.
Is there any quick way in which I can move the Structure, Function Module, Business Object and the BAPI code itself from one SAP system to another?
Yes. Use the built-in Change and Transport System (CTS), preferably by creating a Transport of Copies (ToC). You may also want to check this question.