Hmmm. In a civil rights sense, there are two federal laws that might be involved here, Title VII of the Civil Rights Act (which prohibits discrimination on the basis of gender or sex), and the Americans with Disabilities Act ("ADA," which prohibits discrimination against persons with disabilities and also requires an employer to make a reasonable accommodation to make it possible for a qualified employee with a disability to do the job).
Honestly, you can forget about Title VII here. Men and women aren't being treated any different. Moreover, none of the other women object to the vests. This isn't sex or gender discrimination.
ADA on the other hand might give you something to work with. In order to make the case work, you will have to establish that your gender issues constitute a "disability" within the meaning of the law. You may be able to get a therapist or psychiatrist to back you up on this with a diagnosis of some recognized medical condition. (Gender dysphoria? Gender Identity Disorder? Something like that.)
If you can show that wearing the vest triggers gender-related stress or anxiety that interferes with your ability to perform the duties of your position, ADA may require the employer to make a reasonable accommodation. You have to make request to the employer for a reasonable accommodation. (Sounds like you have already done this.)
What you would like is for the employer to excuse you from wearing the vest. But it's not that simple. The employer is not necessarily required to give you the accommodation that YOU prefer. ADA only requires the employer to make "a" reasonable accommodation. Often, the employer ends up with final say over what accommodation is reasonable.
Once you have requested reasonable accommodation, the employer is required to engage in an "interactive process" of discussion with you to determine whether you need an accommodation, and if so, what accommodation is appropriate.
Here's a thought. (Don't know whether this would be realistic in your case?) A large facility like a megastore might have enough of a receiving/back stockroom operation that it would be possible to reassign you to work entirely in the stockroom - without direct customer contact. In that case, the employer could excuse you from wearing the vest without compromising its own policy about uniform appearance of employees on on the sales floor. Would that work?
What I've written above is based on US federal law, which applies to people in all 50 states. You may also have similar protections (or even greater protections) under Minnesota law.