Well, there's good news and bad news. The (sort-of) good news is that there is the UF_UI C API library that allows you to call several built-in dialogs like File-New (UF_UI_create_part) or File-Open (UF_UI_open_part), as well as some feature selection and file selection dialogs (I'm sure the other APIs have similar functions, like NXOpen.FileNew etc). The bad news is that the "Add Component" dialog specifically is not among them, meaning that if you want to let users add components to their assemblies via a dialog window you have to do the following:
1) Either construct the dialog yourself with Block UI Styler (complicated, since you'll also have to implement the subsequent Constraints dialog);
2) Or call a macro file from your program that clicks the necessary button (also complicated and mostly futile, since macros are very environment-dependent, not to mention obsolete);
3) Or complain to Siemens PLM until they implement the missing function (good luck with that).