Planet Organic on Quinpool Road (6485) used to be locally-owned under a different name, but now it's part of a national chain. It's still pretty good, though, for more unusual convenience products if that's what you're after. It offers a much better bulk, spice, and vitamins section than either of the big groceries, but it's expensive. If you're just after a box or two of Annie's (or something else widely available) you're better off at Atlantic Superstore.
Whole Foods does not have a store here. Both of the major grocery chains (Atlantic Superstore and Sobeys) have organic sections. Your best bet may be Pete's Frootique (2 locations; Bedford's Sunnyside Mall and downtown Halifax in City Center Atlantic just off Spring Garden on Brenton St.) which is a nice place for produce and higher-end groceries. There are also a few "health food" stores if you are more into that genre -- probably your best bet on those is to check the Yellow Pages upon arrival.