Does the vendors’ storage method affect shellfish quality?
Temperature is the most important factor in shellfish storage.

Also, keeping shellfish in fresh water (compared to sea water) is not a good thing.

From the British Columbia Center for Disease Control  (pdf)

"Do not put live shellfish in a closed container or into fresh water (the shellfish will suffocate and die)".

Anecdotal, most of the shellfish I've seen in either supermarkets or in specialty stores are either directly on ice or in pans over ice; but never under running water or in water.