If my children were involved in a car accident with their dad, canI sue the at fault driver on their behalf?

When you say, "His insurance policy lapsed...", Are you referring to the at-fault driver or your childrens' father? 

If the at-fault driver had auto insurance, a personal injury claim for each child can be filed with the at-fault driver's insurance carrier.

If the at-fault driver did not have insurance, but your childrens' father had uninsured motorist coverage on his policy, the personal injury claims of your children can be filed with their father's auto insurance.

If the at-fault driver did not have auto insurance and the childrens' father did not have uninsured motorist coverage, then the recourse would be to sue the at-fault driver for negligence on behalf of your children.  You will need to be appointed guardian ad litem to file a lawsuit on behalf of your children since minors cannot file a lawsuit.

It would be premature to file a lawsuit until your children complete their medical treatment and are released by the doctor or are declared to be permanent and stationary which means reaching a point where no further improvement is anticipated so that you will know each child's total medical bills and have each child's final medical report.  When each child completes his/her medical treatment and is released by the doctor or is declared to be permanent and stationary, obtain the medical bills and medical reports.  Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of each child's injuries and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.

If the at-fault driver has auto insurance or if not if the father has uninsured motorist coverage, file each child's personal injury claim with the at-fault driver's insurance or if no insurance with the father's insurance company as an uninsured motorist claim.  The personal injury claims will consist of the medical bills and medical reports.  Each child's claim is handled separately.  If you and the father of your children are dissatisfied with settlement offers from the insurance company, you can reject the settlement offers and sue the at-fault driver for negligence.  If the case(s) is/are settled with the insurance company, NO lawsuit is filed.  If the case(s) is/are NOT settled with the insurance company, your lawsuit for negligence against the at-fault driver and registered owner if the registered owner is someone other than the driver, must be filed prior to the expiration of the applicable statute of limitations or your children will lose their rights forever in the matter.  Your lawsuit would only be on behalf of the children whose cases were not settled with the insurance company.  You would file one lawsuit naming the children as plaintiffs.  Again, you need to be appointed guardian ad litem to file a lawsuit on behalf of your children.