First woman almost to register as competitive rider (7)
I believe the answer is:
eventer
'competitive rider' is the definition.
(I've seen this before)
'first woman almost to register' is the wordplay.
'first woman' becomes 'eve' (first woman in the bible).
'almost' means to remove the first letter.
'to register' becomes 'enter' (entering is a kind of registering).
'enter' with its first letter taken away is 'nter'.
'eve'+'nter'='EVENTER'
'as' acts as a link.
(Other definitions for eventer that I've seen before include "competitive horse" , "Equestrian competitor" , "Equestrian horse" , "Rider in contests" , "Jumper" .)