At writeLaTeX.com, we compile automatically, typically every 5s-10s. One of the things that makes it work is that the server can cache most of the stuff that doesn't change often, such as the figures. This does require a bit more sophistication on the client end than just uploading all the inputs for every compile, but it's definitely possible. In terms of packages, we run a full TeXLive distribution on the back end, which covers most things. (But we still have to install CTAN packages on request from time to time.)

As mentioned on HN (https://news.ycombinator.com/item?id=5537273), this is something we'd be happy to support on writeLaTeX (https://www.writelatex.com). I should also point out that there is a fairly well-established protocol for online latex compilers, namely that of the CLSI (https://github.com/scribtex/clsi), which is maintained by jpallen at ShareLaTeX. So, it should be straightforward to do this in a vendor-neutral way.

I'm one of the developers. Your speculation is correct! but we are working on a converter. Thanks for the feedback :)

I'm one of the developers. We use pdflatex on the back end, so you're right: eps files aren't yet supported, but we are working on a converter. Thanks for the feedback :)