The chap wearing flat hat is vulgar (5)
I believe the answer is:
cheap
'vulgar' is the definition.
('cheap' can be a synonym of 'vulgar')
'the chap wearing flat hat' is the wordplay.
'the chap' becomes 'he'.
'wearing' means one lot of letters goes inside another.
'flat hat' becomes 'cap' (I've seen this before).
'he' put inside 'cap' is 'CHEAP'.
'is' is the link.
(Other definitions for cheap that I've seen before include "Economical - rubbishy" , "Inexpensive or tastelessy showy" , "Tawdry - inexpensive" , "A snip" , "Inexpensive or tasteless" .)