Requiring an ID for an American to enjoy a Constitutional right guaranteed in the Bill of Rights is blatantly unconstitutional.
Do you want to rent or purchase a car? You need a driver's license or other state Identification.
Do you want to get on a commercial airplane? You need a driver's license or other state identification and passport if traveling outside the country.
Do you want to book passage on a cruise line? You not only need a driver's license of other state identification, you must present a passport.
Do you want to purchase a house? You must present proper identification.
Do you want to stay in a nice hotel? You need a driver's license or other state identification.
Do you want to access your bank account via a bank-teller? You need to present proper identification.
Do you want to rent equipment from a rental store? You need to present proper identification.
The list goes on, but you get the idea.
The framers of the US Constitution and its Bill of Rights set it up so that anything not covered in the Constitution, falls under the purview of the states. The framers of the Constitution, cited in the document that, in essence: The freedom of speech, shall not be abridged (meaning shortened or curtailed). However, we do have laws restricting the freedom of speech.
In the case of the Second Amendment, the framers made it clear that in order for our civilian citizens to help protect their state from a tyrannical foreign....OR....domestic government, the civilians can keep and bear (carry) arms. It did not however, lay down any clarification with regards as to ensuring any individual(s) are not a threat to the state or public. As such, just as I have said, anything "not" covered in the Constitution, falls under the purview of the individual states and so, any state requiring proper identification to purchase a firearm, is within their right to set the local laws accordingly on such matters.