Your application should work without installting Delphi if it wasn't built to use runtime packages(BPL files).
If it was built to use runtime packages, you'll need to deploy them with your application. Since all delphi's runtime package are installed with Delphi, that may give the impression that installing delphi is required for the application to run, which is not the case.
You might however need to install delphi to retrieve the required BPL files.
BPL files are special DLL files that delphi can refers to instead of linking all code into the .Exe, making EXE files much smaller.
So if you manage to get the BDE up, create the right aliases in it, and get all required BPL(if any), the application should start up correctly.
1 last thing that might need to be deployed with your application is midas.dll. (If the application use, for exemple, Tclientdataset)
That's all I can think of for now...