No, you do not need to submit any proof for the exemption for "coverage was unaffordable".
The exemption code is on Form 8965, which is part of your tax return, but you don't need to submit anything else besides Form 8965.
However, be aware that most people do not qualify for that exemption, so be sure that you actually qualify for it (if you have any questions about it, please post back here).
Because most people do not qualify for it, it is quite possible that the IRS will ask you to prove it at some point in the future, so be sure to keep all of your documentation proving that you are eligible for it.
