As far as I can tell, any epub reader that has an affiliated bookshop, will let you buy books right from the ebook reader itself (just like the Kindle does with Amazon) - no actual need to go through a computer, download with ADE and then transfer the file to the reader.
It is of course possible that the neighbour is reading something completely obscure, sold by some minor publisher directly (in which case it's likely not on Amazon either, so "epub reader" shouldn't be the issue), but in my experience small publishers/authors who only sell directly often sell DRM-free files anyway, if only because that's the only way they can sell Kindle-compatible files.
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