Hi Mike,
Though not technically a museum site (but based within the British Museum), you can get rss results from the Celtic Coin Index (http://www.finds.org.uk/CCI/), all searches can return rss or geoRSS and if you have login rights you can get JSON, XML, KML or CSV. I’ve also tried to build an api for people to play around, but I didn’t have any funding to do everything I wanted. There’s some interesting reuse of the data pulled from this within Sean Gillies’ Mush application and displayed in Google maps here – http://tinyurl.com/yqalcp
We introduced RSS search results from the Scheme’s database (www.findsdatabase.org.uk) in 2004 and many researchers use the facility to alert themselves to when a new object has been recorded that meets their parameters. For example a silver denarius of Hadrian found on the Isle of Wight.
Dan