You should be able to determine who the hosting company is, and offer to pay to keep the site up.

If the hosting company is paid they will make and keep a backup for you but under the permissions/access of the original owner.

If the hosting company gets permission to add you as admin to the site from the site owner, who may not be in touch with you, but may respond to the hosting company, then, (since you are paying the hosting company they will be happy to keep you around) you are home free.