If blockchain, the technology behind cryptocurrencies like Bitcoin, takes off, it could move peer-to-peer “employment” to the next level, making it easier for workers and clients to exchange money and credentialing information without a middleman.

The traditional idea of a nine-to-five office job is already being challenged by the increasing number of Americans who are taking up freelancing for companies like Uber, TaskRabbit and Upwork. About 36 percent of the U.S. workforce is freelancing, according to a recent survey by...