The New York Times ran an article recently on art museum web sites. There are, indeed, some really good ones out there. The Met and MoMA both have really good sites.
I’ve occasionally tried to find work in the IT department of an art museum, or at a company that did contract work on a museum web site or mobile app. It’s the kind of thing I think I would really enjoy working on. Alas, I’ve never gotten a response to any of the resumes I’ve sent out. I did get a callback about a volunteer opportunity at a museum once, but they only wanted people available on weekdays, during normal business hours. And I once knew a guy who was responsible for servicing the phone system at the Museum of Natural History, but I don’t think that’s going to help me.