Josh is a photographer and does Instagram Tips on Twitter (@InstaTips). He'd like to start caching websites so that he can browse through them more efficiently. He'd also like to be able to scape images from them when people submit them.
Leo says that it largely depends on the terms of service. Ink361 has a custom, approved API with Instagram that will do it. There are some good "site suckers". There's Site Sucker for OS X, and HTTrack, an open source solution for Windows.
He should be sure he doesn't download these sites too fast or it could be misinterpreted as a denial of service attack. Leo also says the best thing is to contact Ink361 about their API and see if they can provide a solution.