My children's father has passed away and my current husband wants to adopt them. Is this possible and how would we go about doing it.

Should be a snap. You file for adoption, I recommend getting an attorney to do so. You present the fathers death certificate as proof where you need to get his approval.

And if your receiving ssi for the child/children be sure to notify them so it can be stopped as soon as the adoption goes through.

Yes, you will need to show a death certificate. Contact an adoption agency or experienced family law or adoption attorney as your next step.
All the best,
Michele