Former ACME QA Tester.
The sim plest answer:(Though to be fair, I wonder why exalts can't learn mage style magic, since "making the universe cry while you steal its lunch" is sort of an exalted thing to do. My assumption is that it's mainly to keep some semblence of balance).
Exalted like all other sueprnaturals are awakened.
That means you can awaken your Avatar, because it already Awaken.
They can't Awaken same way a Vampire and or Fera can't Awaken.
You could :
a) Try to go Sam Heigh way and created an artifact that allows you to use True Magic (which is in bad taste).
b) Go tinkering in your or someone else's soul (which might work, but if it doesn;t the results wil at best be gilgul).
Even simpler answer:
Exalted lack an Avatar.
The process of Exaltation frees the Avatar in Cycle, leaving the person alive (and teh Shard takes the palce of the Avatar).
Solution are still the same just that b includes stealing an Avatyar and grafting it on yourself or another Exalted.