./extensions/VoteNY/VoteClass.php line 182 ff. generate the following query

SELECT vote_value FROM "Vote" WHERE vote_page_id = '0' AND username = 'Admin' LIMIT 1

...

CREATE TABLE vote (

I suspect the problem is with the forced ucfirst, not the search path. If you declare the table as >vote<, then a query for >"Vote"< is not going to match. The extension needs to either not duble quote the table name (preferred), or change the schema to always use "Vote" as well.