I don’t think you need a PIVOT for this — just create 12 subqueries, one for each month:
SELECT
Vessel,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=1) as Jan,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=2) as Feb,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=3) as Mar,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=4) as Apr,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=5) as May,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=6) as Jun,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=7) as Jul,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=8) as Aug,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=9) as Sep,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=10) as Oct,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=11) as Nov,
(SELECT TOP 1 Inspector FROM Stuff WHERE Vessel=Tbl.Vessel AND MONTH(Date)=12) as Dec
FROM (
SELECT DISTINCT(Vessel) FROM Stuff
) Tbl
You can make this work for any particular year by adding a AND YEAR(Date)=2010
(for example) to the end of each of the 12 subqueries.