As a general matter, no--the law does not require businesses to have insurance. There are certain exceptions--if a business has vehicles, it has to have auto insurance, the same as any other car owner must; certain industries might be obligated to carry insurance (e.g. common carriers)--but as a general matter, while it is, quite frankly, stupid to not have insurance, it is not required. I am not aware of any laws requiring insurance for airsoft/paint ball. You should not frequent a field that does not have insurance, since if you are injured, if the business is a limited liability company or corporation, the owner(s) are not financially responsible--so if you were hurt and sued, you could only recover whatever money happens to be in the business itself, and if they had drained the funds out of it, you  would have no recourse or recovery.