The Fletcher Free library does obituary look ups in the Burlington Free Press for a charge of $2.00 per search for obituaries after January 1, 1939. For them to do the search, they do expect a date of death or have it narrowed down at least to the week.
To make a request, you may write to:
Fletcher Free Library
235 College Street
Burlington, VT 05401
I recommend visiting their website for more information about their obituary look up policies.