You get it when you pass the Forgery Detection training with the assistant curator.
In the supply room
Don't ask me, o_o.
finish the tests and he will give u a key.
The same way you got in to get the picture piece : use your key.
you can with the key!
Apply for the job with the assistant curator and when you pass the Forgery Detection class, he will give you a key to the Supply Room.
Once you pass the test by the assistant curator, and finish the Forgery training, he will give you a key to the Supply Room.
Once you pass the test by the assistant curator, and complete the Forgery Training, you are hired and receive a key to the Supply Room.
You have no way of going to the Supply Room until you are given the key by the assistant curator. (Complete the Forgery Training.)
you finish getting your job and when you finish he gives it to you
If you complete the rearranging of the paintings, don't wander off until you have finished the Forgery Detection class, after which you get the Supply Room key.
Don't run off to the Supply Room right away. You only get the key after you complete the "Forgery Detection Training". (see related question)