Short Answer: No
Working for an e-Verified Employer is Mandatory for STEM OPT.
For your initial 12-Month OPT you can work for any employer. After completing 1 year and when you apply for STEM OPT Extension, your employer must be an E-Verified.
If your employer is not enrolled in E-Verify, you can request them to enroll in E-Verify and it’s a pretty straight-forward process.
WARNING: While submitting the training plan to your DSO for STEM OPT application, your Employer should be e-Verified. So, plan ahead.