Yes, because because the International Confederation of Wizards decrees that all wizards have the right to wands.
American wizards would have the right to carry wands, but not specifically because they are American. Under international wizarding law, wizards worldwide are given this right. The International Confederation of Wizards has guaranteed wizards the right to carry their wands at all times since 1692.
“2 The right to carry a wand at all times was established by the International Confederation of Wizards in 1692, when Muggle persecution was at its height and the wizards were planning their retreat into hiding.”
- Quidditch Through the Ages
So yes, American wizards would have the right to carry a wand, as wizards worldwide do.
Any further laws on American wizards’ wands would be up to MACUSA.
MACUSA didn’t interact with the No-Maj government, and it’s highly unlikely they’d consider a Muggle law on guns to mean they must allow all wizards the right to wands. MACUSA was entirely separate from the No-Maj government, and likely wouldn’t consider certain laws applicable to the wizarding community. Owning a wand is very different to owning a gun, and they’d be more interested in controlling wands, since wizards misusing them could break the Statute of Secrecy.
In the Old World, there had always been a degree of covert cooperation and communication between No-Maj governments and their magical counterparts. In America, MACUSA acted totally independently of the No-Maj government.
- Rappaport’s Law, History of Magic in North America (Pottermore)
The Second Amendment would therefore have no bearing on MACUSA, and would have no effect on American wizards’ rights to own wands. It’s a different situation controlled by a different body.
American wizards need wand permits - which may be revocable.
Wand ownership was actually more restricted in America. All wizards were required by law to have a wand permit. Presumably, if they don’t have one, it’ll be illegal for them to own and use a wand. In addition, it seems likely that their wand permit could be revoked if MACUSA considered them to be misusing their wand. Wand ownership doesn’t seem to be considered a right by MACUSA.
Legislation introduced at the end of the nineteenth century meant that every member of the magical community in America was required to carry a ‘wand permit’, a measure that was intended to keep tabs on all magical activity and identify the perpetrators by their wands.
- 1920s Wizarding America, History of Magic in North America (Pottermore)
MACUSA seems to control wands much more than the British Ministry of Magic, and owning and using a wand seems to be treated more as an earned privilege (that can be revoked) than a right.