This is tricky since you have to distribute the form with the date field field empty.
If all users will have Acrobat, you could have a script that disables the function to fill-in the date.
For Reader, you need to test the field for the date to be empty and if it is then populate that field.
All users have acrobat. Viewers have Reader, but they don't fill or edit the form. How do I disable the function after it runs once?