Yes. In fact, immigration policy was recently updated in light of backlogs due to COVID that E2 spouses can work "incident to status" meaning they do not need to apply for a separate work permit after receiving E2 status. For example, if an E2 spouses is admitted to the United States with an E-2 Visa, and receives a stamp in their passport and online I-94, that spouse can automatically apply for a social security number and work legally in the US.