Hi Steve,
I believe that HotShoe well summarized all the elements to take in account.
Here I try to be more detailed only to help you better focusing on how to reach your goal.
Fundamentally your architecture is based on three elements:
1. a server (i.e. the PC running the existing DB-aware programs); from your writing I suppose that's a Windows machine
2. a communication medium (Internet, LAN, WiFi, Bluetooth, USB..)
3. a client (i.e. your Android device).
Point 3 is very easy; just design a nice UI and let the user tap a button, select from a listview or any other visual control to send the appropriate command to the server.
Point 2 depends on the way you're going to use the system and that will influence what you've to deploy on the server.
Assuming the Internet as the medium, you may want to have a look at the HttpUtils2 library in order to send commands through the use of GET or PUT to a listening server.
This mandates the server to be publicly reachable (i.e. assigned a public IP, eventualy using services like DynDNS).
On the server you may deploy a traditional webserver+PHP interpreter like Apache+PHP or IIS+PHP where the role of the PHP would be that of listening for commands from the client and executing the desidered local (to the server) program by means of a shell instance.
Otherwise you may easily write some code with the superb B4J (so to do everything in BASIC) to do essentially the same thing. You may find a lot of complete examples in the B4J dedicated area on this excellent forum.
Point 1, as said, depends on point 2 above. Essentially you will deploy a "listening box" and issue shell commands to execute local programs.
As HotShoe suggested, if those programs are already web-aware you may want to directly interface with them so all the picture gets simplified.
udg