Way Republican party plugs American’s celebrity status (7)
I believe the answer is:
stardom
'celebrity status' is the definition.
(being a star)
'way republican party plugs american's' is the wordplay.
'way' becomes 'st' (abbreviation for street).
'republican' becomes 'r' (abbreviation).
'party' becomes 'do' (do is a kind of party).
'plugs' is an insertion indicator (plug can mean to fill a gap).
'american' becomes 'Am' (abbreviation in some contexts).
'r'+'do'='rdo'
'rdo' placed into 'am' is 'ardom'.
'st'+'ardom'='STARDOM'
(Other definitions for stardom that I've seen before include "Great celebrity" , "(The prestige of) celebrity" , "Great fame" , "Celebrity status" , "Status of a top actor" .)