In my experience, you can begin granting medals when you have at least 5 of one type (if you've got 5 trust medals, you can grant trust medals, but not other types). At this low level, you can only grant to non-realmmates though. At a higher amount (10? higher?), you can also grant medals to realmmates. Looking at your family page, you only have one medal total, so that would explain why you can't grant any medals.
Recently, code has been implemented to replenish your medal giving supply over time. Previously, if you ran out of medals to give you had to wait until you were given new medals.