There are different meanings of brace. Erik's answer is perfect for orthodontia. Other common types of braces might be a knee brace or a neck brace. In those cases, you could still use the verb "wear".
He's been wearing a neck brace since the car crash.
But you also could use the word "use".
I wear/use a knee brace when I run.
You would never use "use" for orthodontic braces.