Yes, there are plankton species that are only found in Arctic waters. For instance, there are many species of copepods, and they are usually specific to different regions- for instance, you will find species of cold water copepods in places like the Arctic that you probably won't see in places with warmer waters. Each species has their own identifying characteristics. For some pictures and references, try these two websites:
This one will take you to a page of pictures and names of plankton living in the Barents Sea, which is the eastern equivalent of the Bering Sea: www.nodc.noaa.gov/OC5/BARPLANK/WWW/HTML/bioatlas.html
This one will take you to the other website linked to the BEST cruise in the Bering Sea, and you should be able to find some plankton information on there as well: www.pmel.noaa.gov/foci/ice07
This website talks a little bit about the food web in the Arctic Ocean:
You can also do a google search for "Bering Sea plankton"
Good luck, thanks for the great question!