Arrive with dog, say, to take part (7)
I believe the answer is:
compete
'to take part' is the definition.
'arrive with dog' is the wordplay.
'arrive' becomes 'come' (synonyms).
'with' indicates putting letters inside.
'dog' becomes 'pet' (dogs are often kept as pets).
'come' going around 'pet' is 'COMPETE'.
'say' is the link.
(Other definitions for compete that I've seen before include "Take part in contest" , "Struggle for victory" , "Contend, vie" , "Engage in a contest" , "Take part (in race)" .)