As far as I know, the leading case on the matter is Hale v. Henkel, 201 US 43. There, the court explains
While an individual may lawfully refuse to answer incriminating
questions unless protected by an immunity statute, a corporation is a
creature of the State, and there is a reserved right in the
legislature to investigate its contracts and find out whether it has
exceeded its powers. There is a clear distinction between an
individual and a corporation, and the latter, being a creature of the
State, has not the constitutional right to refuse to submit its books
and papers for an examination at the suit of the State; and an officer
of a corporation which is charged with criminal violation of a statute
cannot plead the criminality of the corporation as a refusal to
produce its books.
The court specifically denies that corporations have 5th Amendment rights:
The benefits of the Fifth Amendment are exclusively for a witness
compelled to testify against himself in a criminal case, and he cannot
set them up on behalf of any other person or individual, or of a
corporation of which he is an officer or employe.
This contrast with protection against unreasonable searched of corporations, per the 4th Amendment:
A corporation is but an association of individuals with a distinct
name and legal entity, and, in organizing itself as a collective body,
it waives no appropriate constitutional immunities, and, although it
cannot refuse to produce its books and papers, it is entitled to
immunity under the Fourth Amendment against unreasonable searches and
seizures, and, where an examination of its books is not authorized by
an act of Congress, a subpoena duces tecum requiring the production of
practically all of its books and papers is as indefensible as a search
warrant would be if couched in similar terms.
Similarly, in Wilson v. United States, 221 U.S. 361,
the constitutional privilege against testifying against himself cannot
be raised for his personal benefit by an officer of the corporation
having the documents in his possession.
An officer of a corporation cannot refuse to produce documents of a
corporation on the ground that they would incriminate him simply
because he himself wrote or signed them, and this even if indictments
are pending against him.
Likewise, United States v. White, 322 U.S. 694 ("The constitutional privilege against self-incrimination is essentially a personal one, applying only to natural individuals") and Bellis v. United States, 417 U.S. 85:
Fifth Amendment privilege against self-incrimination held not
available to member of dissolved law partnership who had been
subpoenaed by a grand jury to produce the partnership's financial
books and records, since the partnership, though small, had an
institutional identity and petitioner held the records in a
representative, not a personal, capacity. The privilege is "limited to
its historic function of protecting only the natural individual from
compulsory incrimination through his own testimony or personal
So only natural persons can plead the fifth.