Your answer will most likely lie in your lease. Others' experiences might not matter as much as their leases are different. What options does your lease give you?
Most mortgages say the lender has the right to buy homeowner's insurance for you if you don't prove you have it. Does your lease allow you to buy renter's insurance and charge the tenant?
My leases say I can charge my tenants anytime I do something or pay for something they are responsible for. I haven't had a renter's insurance issue, but have done this for maintenance items. You might be able to go this route if your lease says something to that effect.
Without lease support, you can write them a letter informing them they are in violation of the terms of their lease due to the lack of proof of insurance. Tell them they have until [Date] to show you proof of insurance or move out of the property. You can add that you will buy the insurance for them and charge them if they remain in the property after [Date]. Alternatively, you can tell them you will charge their security deposit for it, but that becomes a charge against your security for the property.
You should consider whether you really want them out of the property with regards to the above paragraph. You can drop the move-out part and just go with the buy-for part if you don't want to push them out. Otherwise, you can go the other way if you want them out. If it were me with my tenants, I would go with the buy-for option and just charge them.
Next time, update your lease and don't let them have the keys until they show you proof of insurance.