FME Server Workspace
An FME workspace can handle any amount of data in any format if it is predefined in the workspace. If it cannot be predefined then FME has special functionality to read any format of data (what we call "Generic") and read any data schema ("Dynamic").
In this case it's unlikely that the Generic functionality will help. Firstly there is no need for it because the format will always be the same and secondly (as pointed out) the Generic reader won't work with database formats (it's a file only technology).
Whether you will need to use the dynamic functionality depends on the nature of the other Geodatabases.
If all Geodatabases will contain the same set of tables/classes, then you don't need to use the Dynamic tools. You just need to pick a different Geodatabase at run time and it will run to completion.
If the Geodatabases will contain different tables/classes, or the same tables with different attributes, then you need to use Dynamic functionality.
Do it like this:
1) When creating the workspace, check the Dynamic Option on the dialog:
2) This will create a workspace like this, with one layer only and no attributes:
3) Run this workspace (File > Prompt and Run) and you get prompted to select which tables (Feature Types) to read. You can choose any of them:
4) This gives you the required output:
5) The next time you run the translation you can choose a different Geodatabase. This gives a different set of tables to read:
6) But because the workspace is dynamic, it works just the same:
This is obviously being run from within Workbench. But run from Server the effect will be the same.
Hope this helps. Sorry for such a lengthy post!