An "awarder" would be the giver of an award. An awardee would be the "receiver" of an award.

No :) This is an example of a fairly common pattern distinguishing between the one who does something and the one who has something done to them. Here the "awarder" is the one giving the award, to the "awardee". -ee in particular is often user to identify the one who has something done to them, or who is the recipient of an action. Another example might me "Licenser" (giver) and "Licensee" (receiver / holder). HTH