I have accounts with both Emigrant and ING. Orignally I started with ING, but when I learned of Emigrant and their slightly higher yield, I moved most of my money over there, but kept a bit in the ING account, just to keep it open if I ever needed it.
So far, so good. I've never had a problem transferring funds into or out of the Emigrant account (or the ING account, for that matter) The ING site does seem like it's a bit more secure, at least the log-in page. I've had both accounts for over a year now, and I'm satisfied.
However, the concern over a web-based account, versus one at a brick-and-mortar bank, is understandable. The first time I tried something like that was with Signet Bank (I dunno who they've merged and re-merged with since then) back around 1996 or so, only it was phone and mail-based instead of internet, but still not face-to-face with a teller. I didn't like it back then, but I grew into it, I guess.