She promised a reward no questions asked. Pay up.

I'm no lawyer, but it does seem clear cut.

"Promised" how?

Gaga should give her the money and immediately start a civil suit for the woman being involved in stealing the dog. It'd probably pan out even, except for the tons of money the lawyers would skim off it.

Don't make her redognap the dogs

A post on social media isn't a contract.

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So can I steal your car and collect the reward for its return? Or would that be unethical and unlawful?

You can.

If I said return it with no questions asked, then I care more about the car then the punishment.