Can braces make you feel sick after they've been put on?

The only logical thing about braces making you unwell is the discomfort they will cause. That is because suddenly there is an unnatural presences in your mouth of steel brackets and wires. They may make you more aware of your teeth than before. The tightness of the wire to pull/push your teeth into shape is also going to make you feel uncomfortable.
However anything that has been used in the procedure or been placed in your mouth, has got to be approved and tested. That is why it is unlikely that braces are making you sick.