If an employee of my LLC negligently causes injury to me in the course of their employment, can I sue my LLC, like a non-owner could?
YesUnless your business is a sole proprietorship it operates as a legal entity seperate from its owners.
It owns its assets an acrues its own liabilities. It can be sued and it can sue others. It can also agree to its own contracts.
Typically the only thing a legal entity that is not a natural person cannot do is sign a marriage contract.
Depending on the industry there may be no need to sue. There exist many mandatory workplace insurance to cover accidents in the workplace.
You may have to share details of the injury with them but they may be uninterested in whos at fault.
You and the company you own are separate entities. That’s why yo can’t just spend your company’s money, you have to receive it as salary or dividends. Usually.
If your company causes anyone damages of say £20,000 the company can and should pay the damage. Same if the company causes damages to you personally. Obviously you personally would have the damages and the compensation and therefore zero profits. The company would have £20,000 less cash, £20,000 less profits to pay taxes on saving £4,000 and therefore £16,000 less money to pay you or other owners dividends in the future.
Suing should of course be unnecessary if you are the sole owner. Tim Cook might have to sue if an Apple employee dropped a Mac onto his car. There might be legal problems if you as the owner ordered the company to do things that are objectively bad for the company, like in the £20,000 case pay a £100,000 settlement. (Actually, if you have insurance paying for the damage, they might disagree with any payment.)