As vclaw said you can use OpenStreetMap Data to find postboxes.
An easy way to achieve this would be to use the Overpass API to download a GPX file and to add this to you GPSr.
I prepared a sample query for Vienna and postboxes here. As you can see, you can export the data in different formats, one of them is GPX. Change the coordinates for other cities or the whole EU and maybe try to add post offices if you like.
There are a lot of postboxes and post offices mapped in OpenStreetMap. Coverage and accuracy may vary, but I think it is generally pretty good for most of Western Europe. Considering just how many postboxes there are, and how spread out they can be in rural areas, there will be some missing from OSM.
Depending on what sort of GPS device you are using, you may be able to load OpenStreetMap based maps onto it. eg you can do this with most Garmin models. Then that would let you search for your nearest postbox or post office. Or if you using a smartphone, there are a variety of apps using OpenStreetMap, some have options for searching for points of interest and offline mapping if you want. Or you can just download points for the postboxes, and use these as an overlay for another map. How to do this will depend on what device you are using, and what data format it requires.
Not sure about mailboxes themselves, but post offices, yes.
Geonames is free, and in its datasets of POIs, check out the list of feature codes, specifically:
S.PO post office a public building in which mail is received, sorted
and distributed
Most post offices have mailboxes, so even if it doesn’t cover every street corner mailbox, it’s probably a very good starting point.
The data may require some manipulating to get it into your GPS device though…
Credit:stackoverflow.com‘