I've been on one of the boat-based trips to Svalbard (this was some years ago when there was a fleet of small (50 berths) Russian-owned ice-strengthened ships doing the rounds). Ours was in late June. This is fairly early in the season, but supposedly best for polar bear. There was still lots of ice everywhere. A month later there should be less of that, and presumably young birds will be around.
Are you going to be on a boat, in which case I assume you're going to follow the west coast north from Longyearbyen, or are you going to be land-based? This will make some difference to what you are likely to see. We found Red-throated Diver (on lochs), Pink-footed and Barnacle Geese as well as Snow Bunting relatively common on land. We got very good views of Rock Ptarmigan, and found some very confiding Red/Grey Phalaropes.
Near Longyearbyen we also had breeding Eiders, Arctic Terns and Ivory Gull.
If you're on the water you can expect to see lots of Fulmars, all Skuas (bit of luck needed for Pomarine), King Eider and Long-tailed Ducks. Purple Sandpiper is reasonably common on rocky shores. You need to be lucky for Sabine's Gull (I think there's a known island with some breeding birds). Glaucous and Black-legged Kittiwakes are abundant, as are Bruennich's and Black Guillemot, Little Auk and Atlantic Puffin.
Mammal-wise you should get the local form of Reindeer, and Polar Fox is fairly easy. Walrus if you go to the right places, as well as Ringed and Bearded Seal. We also had Beluga, Minke and Humpback Whale.
Andrea