So I haven't completed the solution, yet, but it looks like you can use this tool:
Conversion > JSON > Features to JSON.
This will convert your shapefile (in my case 81 polygons) to a JSON file. You can open this with a text editor to see that, indeed, all the vertices are listed for each polygon.
Further, the python standard library (import json) treats the json objects as dictionaries. You can then simply loop through your dictionaries to write the vertex values (and any other attributes you want) to a csv file. If I get it to work I'll come back & post the soln.