Recently, Autonomous Vehicles (AVs) have gained extensive attention from both academia and industry. AVs are a complex system composed of many subsystems, making them a typical target for attackers. Therefore, the firmware of the different subsystems needs to be updated to the latest version by the manufacturer to fix bugs and introduce new features, e.g., using security patches. In this paper, we propose a distributed firmware update scheme for the AVs' subsystems, leveraging blockchain and smart contract technology. A consortium blockchain made of different AVs manufacturers is used to ensure the authenticity and integrity of firmware updates. Instead of depending on centralized third parties to distribute the new updates, we enable AVs, namely distributors, to participate in the distribution process and we take advantage of their mobility to guarantee high availability and fast delivery of the updates. To incentivize AVs to distribute the updates, a reward system is established that maintains a credit reputation for each distributor account in the blockchain. A zero-knowledge proof protocol is used to exchange the update in return for a proof of distribution in a trust-less environment. Moreover, we use attribute-based encryption (ABE) scheme to ensure that only authorized AVs will be able to download and use a new update. Our analysis indicates that the additional cryptography primitives and exchanged transactions do not affect the operation of the AVs network. Also, our security analysis demonstrates that our scheme is efficient and secure against different attacks.