You can file a workers' compensation claim, but you cannot sue your employer. See http://law.onecle.com/california/labor/3602.html for the exclusive remedy rule and exceptions (which do not apply to your case). You have a cumulative trauma (http://www.wcwebzine.com/a-z/cumulativetrauma.htm). Fill out the claim form: http://www.dir.ca.gov/dwc/DWCForm1.pdf and submit to your employer. If the claim is denied, retain an attorney.
You an file a workers compensation claim and receive benefits if your injury is caused by work. One of the jobs of the doctor in workers compensation is to determine if your injuries are work related or not.
I would recommend that you give your employer a claim form and they should send you to a doctor to start your claim. If you run into problems with your claim you can seek the services of an attorney, however your claim may be uncomplicated and an attorney will not be needed.
Here is a link to a claim form. http://www.dir.ca.gov/dwc/dwcform1.pdf
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This is an interesting issue that I've heard posed before at workers compensation conferences. I agree with the others that you can file a claim, but I'd be interested to find out if it was determined "work related." (arising out of the scope of your employment). You don't get compensated for pain and suffering in workers compensation and if your private insurance is covering the treatment, you should ask yourself what you hope to accomplish by filing a claim. I wear heals myself and it can be brutal to the feet, I do question whether it falls under the category of work related. It's an interesting topic and I don't know if you will prevail but I'd be curious to see what happens.
As others have noted, you cannot sue your employer directly under CA workers' compensation laws for a work related injury. There are exceptions to this rule, but with your employer having workers' compensation insurance, one of those exceptions doesn't apply to you.
When you say "file a claim with Workers Compensation Fund" I take it you mean State Compensation Insurance Fund (SCIF)? If so, then that would be the insurance carrier your employer uses for workers' compensation. You would request a DWC-1 form from your employer to complete and return to your employer. That starts the work comp process. Your doctor, if the doctor suggested it was work comp, would then be required to bill and request authorization for further treatment through SCIF. You may have other non-work related issues which are the sole cause or are contributing to your injury.
Workers' compensation is a complex area of the law with several twists and turns that often surprise, to their disadvantage, injured workers. My best advice is to consult an attorney whose practice incorporates workers' compensation, preferably one who is a member of the California Applicants' Attorneys Association (CAAA). Good luck and regards! -Rob Bicego