In MS Word 2003 can an user execute a VBA macro with inside some MsgBoxes and at the same time modify the open word document?
With an human-type example:
Sure, just make the form modeless² when you show it:
UserForm1.Show (False)
This will show the form. Yet, you can still navigate and use the document. Yet, once you press a button within that form the VBA code attached to that button will run.
²You can learn more about modeless at MSDN: https://msdn.microsoft.com/en-us/library/office/gg251819.aspx