I cannot sell my pet to anyone. When I select sell pet, they just chat. Nothing happens. What should I do?

I guess it depends on what the goal is here. If you wish to shift a pet to another household and cannot do so socially, you could use MasterController's Add Sim to force the pet into the other household. Or I believe the same thing could be done through a Split and Merge function on the active household through Edit Town.

The Sell Pet interaction I believe comes from the Consigner mod. Without that the corresponding social interaction would be to give the pet away to a friend or relative. But in order for Sell Pet to work, the animal has to have some value associated with it. Can you sell it at the Consignment Store?
Consigner-FAQ#When-I-try-to-sell-my-pet ... -value-why

I believe you also need to be at least friends with the person you're trying to sell to.