Take part in dog show, perhaps, but arrive without dog? (7)
I believe the answer is:
compete
'take part' is the definition.
'arrive without dog?' is the wordplay.
'arrive' becomes 'come' (synonyms).
'without' is an insertion indicator ('without' can be similar in meaning to 'outside').
'dog?' becomes 'pet' (dogs are often kept as pets).
'come' going around 'pet' is 'COMPETE'.
'in dog show perhaps but' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for compete that I've seen before include "Struggle for victory" , "Take part (in race)" , "Contend (for a prize)" , "Vie, contest" , "Engage in a contest" .)