Latter-day Saints are usually aware that, for ancestors born within the past 110 years, certain rules apply to reserving them for temple ordinance work. If you are not one of the closest living relatives, you need to get permission from one of the closest living relatives and then submit a request to be able to do the work.
Hopefully the reason for this is obvious. We want to give the closest relatives first priority in doing the work for those they knew and loved. Or, if they are not LDS and do not want us to perform temple ordinances for their close relatives, we need to respect that too. The relatives who can give permission are an undivorced spouse, adult children, parents, and siblings of the deceased.
What if there are no living people who fit into those categories? No problem. Go ahead and click the link to request permission and fill out the form. In the section where you would say who gave permission, simply enter "no close living relatives" for the name. For the relationship, select "other" and enter "no living relative" in the box next to it. In the box where you are to enter contact info for the person who gave permission, put in your explanation regarding the close relatives.
Sometimes the folks who approve these requests will contact you and ask for proof that no living family members are living, so be prepared to gather the evidence you can find. Obviously we want to offer all who have ever lived the opportunity to accept or reject the gospel of Jesus Christ, but we also want to respect the desires of close family members, so those who review and approve or reject requests try to be very careful with these cases. Be patient and provide all the information you can to make their jobs easier.