The Issue is Unsettled
In US law, corporations have been held to have some, but not all, of the rights that natural persons have.
Specific Rights and Cases
Corporations have at least limited rights of free speech (see Citizens United v FEC), but not the right to vote or hold elected office. They clearly have the right to sue and be sued, and have had it since the earliest days of the US as a nation.
In Trustees of Dartmouth College v. Woodward, 17 U.S. (4 Wheat.) 518 (1819) the US Supreme Court held that a corporation was entitled to the protection of the Contract Clause of the Federal Constitution against a state law attempting to reorganize the corporation.
In Society for the Propagation of the Gospel in Foreign Parts v. Town of Pawlet (1823) The Supreme Court held that a foreign (British) corporation had the same rights to own land as a natural person.
In Santa Clara County v. Southern Pacific Railroad Company, 118 U.S. 394 (1886)](https://en.wikipedia.org/wiki/Santa_Clara_County_v._Southern_Pacific_Railroad_Co.) Chief Justice Morrison Waite statedat the start of the oral argument that the Supreme Court justices unanimously believed that the Equal Protection Clause did grant constitutional protections to corporations, althoguh this was note stated explicitly in the actual opinion
Corporations do not have the fifth amendment right agaisnt self-incrimination., That was rejected in United States , v. S. Steve Sourapas and Crest Beverage Company, 515 F.2d 295 (9th Cir. 1975) The Nineth Circuit court wrote in that case:
It is firmly established that a corporation has no fifth amendment protection against self-incrimination and that neither the corporation, a corporate officer or any other person can prevent the production for examination of relevant corporate records. Bellis v. United States, 417 U.S. 85, 94 S. Ct. 2179, 40 L. Ed. 2d 678 (1974); United States v. White, 322 U.S. 694, 698, 64 S. Ct. 1248, 88 L. Ed. 1542 (1944).
1 USC 1 states:
In determining the meaning of any Act of Congress, unless the context indicates otherwise—
the words "person" and "whoever" include corporations, companies, associations, firms, partnerships, societies, and joint stock companies, as well as individuals
See also the Wikipedia article "Corporate personhood"
I am not aware of, and hav not been able to find, any US court case in which the issue of whether the second amendment right to "keep and bear arms" applies to a corporate person was addressed and decided. Since the justification for incorporating that right in Heller was the need for personal self-defense, it would not seem to apply to a corporation. But a court might find a different basis.
Until this issue is raised and ruled on in a court case, it must remain unclear in US law.