Shoulder cord

Do I have to fill out a permission form, similar to this one (http://www.capmembers.com/media/cms/F002A_378E56CC9AD27.pdf), to wear a shoulder cord I earned? (I'm a cadet).

Which shoulder cord is that?

Off the top of my head, I can't think of any cords which are earned meaning as a perpetual "award".

Your wing might have something local, but in any case, cords are not awarded or approved via a Form 2a.

Cords are generally an indicator of some office or position, CAC, COTQ, NCC Color Guard Team, member of an Honor Guard,
so the approval would come from Personnel Authorizations, memos, or internal procedures of the echelon it is worn for.

As an example, properly appointed unit CAC Primary reps wear the red cord.   Once properly appointed, you wear it with no
further approval necessary.  When you are relieved, you remove it.

Academic cords are usually worn in the quarter, or the quarter right after, a given level is reached (unless your wing has something special).

In addition to the above, your best option is to ask your Leadership.