tl;dr: On 15 october, we’re going to: drop table PUID;
In 2006 we added support for PUID acoustic fingerprints from MusicIP. MusicIP went out of business some years ago and the PUID service has been passed along, through various hands. Along the way it became neglected and the quality of the service went downhill. This spurred the creation of AcoustID which is our preferred solution for fingerprinting inside MusicBrainz today. We set out to let AcoustID support and PUID support live side-by-side in MusicBrainz for a while and we feel that almost enough time has passed. Therefore we’re going to remove PUID support from MusicBrainz in our autumn schema change release on 15 October, 2013.
If you depend on PUID support today, we encourage you to move over to AcoustID as soon as possible.
MetaBrainz Blog 
Is it possible to search for AcoustIDs as easy as for PUIDs?
(i cannot find any field on http://echoprint.musicbrainz.org/search or the APIs)
You might be looking for this: http://acoustid.org/webservice#lookup_by_trackid