Workers Comp Insurance?
Workers Comp Insurance does exist for the Self Employed.
It is not required by most states for Sole Proprietors, but if you are an LLC or an S-Corp, many states allow the owners to Opt Out of coverage.
But what you are facing is a different issue. By working on someone else's site, you are legally an IC. Thanks to the wonderful world of construction contracting, employers must cover their ICs if the IC does not have proof of WC insurance. In your situation, the "client" is the employer, using you as an IC. They have no interest in covering you under their WC policy, so they are asking you to provide proof of your own.
While you are not legally required in CA to have your own WC, your cient is insisting on it, and it is probably the client's Risk management / legal department reviewing the terms, not your HR or other operating Dept contact. They don't see shades of grey, they just see a contract and all contractors have to be on the same terms. Courts in many states, particularly CA, have made WC awards to ICs. One of the bases for doing so is the repeated use of the IC to do the same job over and over -- if you have that big a need, hire someone to do it as an employee. Using an IC repeatedly confers some level of employee status
It gets worse. You can't just get WC coverage for the work you do at the client requiring it, you will probably have to get enough to cover your entire business. Insurance Companies also don't see shades of grey. Once a year, you have to send in a WC compliance audit that shows the entire payroll for your policy, which means they want to see your W-2. your premium is $x per $1000 of coverage, but you can't pick and choose which $1000s are covered. So you will have to get a WC policy that covers your entire income, to the tune of about 5% (the going rate in CA for a company with no "experience rating". You are likely to spend 5% of your income covering yourself in case you get careless and hurt yoruself on the job. Non-job related injuries are not covered by WC, that is Disability, like AFLAC.