The reason I ask is that there is no data base you can access and no agency you can call that will say, "Oh, yes, your employer has a workers comp policy with XYZ company". Workers comp information does not appear on the OSHA poster but on a separate poster, and from what I see on the state web site that poster is currently under revision by the state. It's quite possible, from what I see there, that your employer is currently transitioning to the revised poster which is why you don't see it at the moment.
The only thing you can do is ask your employer for the information and you have done that.
IF it should happen that you should at some point have a claim AND it then appears that your employer does not have coverage, you will have recourse. But under the circumstances I would not suggest pushing the issue since you actually have no immediate need for the information. Just a suggestion.