Right away, family member offers transport (7)
I believe the answer is:
elation
'transport' is the definition.
(I know that transport can be written as elation)
'right away family member' is the wordplay.
'right' becomes 'r' (common abbreviation).
'away' is a deletion indicator.
'family member' becomes 'relation' (a relation or relative).
'relation' with 'r' taken out is 'ELATION'.
'offers' is the link.
(Other definitions for elation that I've seen before include "Toenail (anag) -- delight" , "Great happiness and exhilaration" , "exhibits successful pride" , "Feeling of delight or pride" , "Feeling of joy and pride" .)