Dozens of startups that want to make the leap from unregulated cryptocurrency dealer to licensed brokerage are waiting for regulators to open the door to let them proceed.

Around 35 to 40 companies involved in digital assets have applied with U.S. securities watchdogs to become brokerages, with many hoping to launch electronic trading platforms, according to the Blockchain Association, an industry group. None of the applications has been approved since regulators stepped up enforcement actions against issuers of cryptocurrencies...