If the employer had worker's compensation insurance at the time of your injury, the insurance company is liable for providing you with benefits. If your employer was self-insured for worker's compensation, and the entire company is being dissolved through bankruptcy, there is a separate fund that would step in to pay your benefits. If your employer was illegally uninsured, you would be able to collect your benefits from the Uninsured Employers' Benefit Trust Fund. In any event, your benefits should not be interrupted or lost due to your employer's business closing, unless part of your claim is for increased benefits that are ordinarily not covered by insurance, such as serious and willful misconduct or discrimination against injured worker.
In Missouri, the employer and the insurer are both parties. Provided the employer paid premiums through the injury date your claim will still be covered by the insurer regardless of the employer's status: unless bankruptcy is filed and then a stay maybe placed on your claim. Consult your local attorney.
Yes, it is the insurance carrier who has responsibility for continuing your coverage. If the employer is self insured and goes out of business there is still coverage for your claim. If you have a question you can always ask the Information and Assistance person in your jurisdiction.