Do I need sponsorship for work permit if my I-526 has been approved?

No. When you applied for your adjustment of status, you should''ve also applied for an Employment Authorization Document ("EAD") which would give you the authorization to work in the U.S. until your permanent residence/adjustment of status is approved.

If you are waiting for adjustment of status, does it mean you have filed your I-485? If yes, you should file the Application for Employment Authorization Document, Form I-765. In this regard, you do not need a sponsor. Advisably, consult your attorney or any attorney for options you may have on this issue.

It appears you can answer no.

The work permit you would get after applying for the adjustment of status is based on your own process, no employer required.

No, you do not need sponsorship but you should have applied for an EAD as part of your adjustment process. You cannot use your F1/OPT EAD once you file I-485. File an EAD application as soon as possible and wait until it arrives before starting work. Work without authorization would prevent you from adjusting status in the U.S.

If you filed your I-485 together with an I-765 (work authorization request), then you can expect your temporary work authorization to arrive approximately 6 months after filing. You will be able to work on your temporary work authorization until the adjudication of your change of status (I-485). Therefore, should not need sponsorship.

If you have OPT, technically you do not need sponsorship in terms of a petition filing to USCIS by your prospective employer. They might have to write a letter and meet other obligations, such as E-Verify, since you are on STEM OPT. They do not have to be your sponsor or petitioner in any application or petition to USCIS as long as you maintain either STEM OPT or EAD based on a pending Form I-485.