AcoustIDs are not included in the MusicBrainz database directly, so they can't be searched using the MusicBrainz Web Service. There is also an AcoustID Web Service.
You can't lookup by acoustID, yet. You can only lookup a fingerprint created by chromaprint: http://api.acoustid.org/v2/lookup?client=8XaBELgH&meta=recordingids&duration=641&fingerprint=AQABz0qUkZK4oOfhL-CPc4e...
There are multiple fingerprints mapped to one AcoustID.
EDIT:
I opened a ticket on MusicBrainz
and a ticket for acoustid.org.
EDIT: What MB calls "AcoustID" is an "AcoustID track ID" on acoustid.org and you can access it via http://api.acoustid.org/v2/lookup?client=8XaBELgH&meta=recordingids&trackid=453a8e75-ef90-4f31-a2b2-84494219bd3b (since 2011 already). There is no documentation about that yet, but there will soon be.
EDIT: This is now documented in http://acoustid.org/webservice#lookup_by_trackid