Create Shopify Private App

We have a company that offers sports facilities rental deals - for example renting a batting cage. We are working on an app to compliment this. Customers will use the app or website to book their reservation at a discount. What we are looking for: To create a Shopify Private App - where the sports facilities are registered and can import their Google Calendar. They should be able to create a new event - only under their facility and for the given products we will create or allow them to use. They will only have to click to add when they are offering their product. We have looked at multi-vendor app and are okay using this if it can be customized or integrated with our UX.