Hi Berm, an API to expose data from your in-house database would be a good first step. But stepping back a little, how do you plan to enable access to your in-house server(s) to outside applications ? If you have a public IP address you can NAT (network address translation) traffic from outside on this IP into your in-house server(s) - this can be http traffic, traffic on specic ports (eg to a database server), socket to socket or whatever traffic you decide to allow. Alternatively (I have no experience of this) some cloud providers sell Virtual Private Networks that can connect to your own network.
Once this connectivity question has been resolved the system design options can then be narrowed down a lot more.