Unfortunately Immigration law does not permit EAD issuance to H4 status, however its only for L2 status till now.
Now being on H4 you are eligible for applying EAD incase you Or your primary applicant (H1) has an approved I140 and/Or has a pending application for adjustment of status more than 180 days (pending for adjucation) ago. And on this I140/ I485 you are mentioned as beneficiery for LPR status. If you meet this conditions, you may apply for your EAD (while the primary applicatnt is still on H1 status) thru online Or paper process to the appropriate place.
Hope this is clear.