Irishman carries duke in chair (5)
I believe the answer is:
sedan
'chair' is the definition.
(sedan chair is a type of chair)
'irishman carries duke' is the wordplay.
'irishman' becomes 'sean' (common Irish name).
'carries' is an insertion indicator (inserted letters carried inside).
'duke' becomes 'd' (abbreviation).
'sean' enclosing 'd' is 'SEDAN'.
'in' is the link.
(Other definitions for sedan that I've seen before include "Once a mobile chair for Danes" , "Palanquin; car" , "Chair; car" , "In cars, an alternative name for 'saloon'" , "A closed litter for one passenger" .)