Most likely yes. As Poseidon made the Horse and Athena the Chariot they together created the horse drawn chariot and it was probably the only thing they did together
Poseidon with his wife Amphitrite
No, they did not do so in myths.
Gods are immortal. Zeus didn't want an unending war with Gods being forced to take sides, and godly carnage destroying his kingdom. So he forced them to sort out conflicts in a civilized way. However, after the Athens incident Poseidon and Athena where always in competition with each other, at Troy, with Odysseus. The only thing the managed to cooperate on was the horse drawn chariot.
In a chariot drawn by goats.
The chariot.
THOR
Every god could have a chariot if they wanted to. Aphrodide sometimes had one drawn by dolphins.
The Horse Drawn Chariot was invented in the year 1,700 B.C.E.
The chariot of Apollo
A chariot drawn by horses.
Chariot
cuniform, wheel, horce-drawn Chariot