If you want to contact supporters to request a Gift Aid Declaration, or simply report on who does not have one, the most reliable approach is to tag constituents who do have a valid GAD, then exclude them from a Constituent List.
Find and Tag GAD constituents
You could also set up an Auto-Tag rule to do this automatically so you don't need to manually add the Tag on a regular basis.
Once all constituent profiles are Tagged accordingly, you will need to exclude them from a Constituent list.
Exclude Constituents from a new list
After the constituents were tagged accordingly:
Create a new Constituent list.
Filter the list by Does not Have any' of <your Tag>.
Optionally you can add additional filters.
The results will be all those constituents that do not have a GAD on their Timeline.