J-2 dependents are eligible to work while in the US and can apply for employment authorization by completing form I-765 from USCIS. Once approved, the J-2 will receive an Employment Authorization Document also known as the EAD card. The J-2 may begin employment only after receiving the EAD and applying for a social security number.
The I-765 application form can be found at www.uscis.gov in the "Forms" area. Further instructions are available under J-2 Employment Instructions, see below.