Under the notion of corporate personhood, can a corporation vote on presidential candidates in the U.S.?
no - corporate personhood is a legal fiction designed to make it easier for legislators and courts. It's what allows you to sue a corporation, what allows them to own property, have them protected under certain amendments, etc. It doesn't mean "Corporations are exactly the same as actual people" it means "Corporations, in some settings, have the laws applied to them similarly to how laws are applied to individual people."
It has been suggested in another answer that it might not be unconstitutional for such laws (allowing corporations to vote) to be passed in the US. I have my doubts, especially at Federal level.
Chief Justice Earl Warren said “Legislators represent people, not trees or acres. Legislators are elected by voters, not farms or cities or economic interests.”
So such corporations-can-vote laws would probably be challenged in courts and baring change of heart of the Supreme Court on the "one person, one vote" principle as referring to actual people... I don't see such laws withstanding a strict scrutiny test.
Imagine that whites in some state registered a lot more corporations than blacks, which would effectively allow a white vote to count more than a black one. That could easily lead to a lawsuit under the Equal Protection Clause of the 14th Amendment. That's exactly the clause that led to Baker v. Carr (1961) (which opened the way for Reynolds v. Sims), because the former set the doctrine that voter over-represention is not merely a political, but a justiciable question:
By the time of Baker's lawsuit, the population had shifted such that his district in Shelby County had about ten times as many residents as some of the rural districts. The votes of rural citizens were overrepresented compared to those of urban citizens. Baker's argument was that this discrepancy was causing him to fail to receive the "equal protection of the laws" required by the Fourteenth Amendment. Defendant Joe Carr was sued in his position as Secretary of State for Tennessee. Carr was not the person who set the district lines – the state legislature had done that – but was sued ex officio as the person who was ultimately responsible for the conduct of elections in the state and for the publication of district maps. [...]
The decision of Baker v. Carr was one of the most wrenching in the Court's history. [..] The opinion was finally handed down in March 1962, nearly a year after it was initially argued. The Court split 6 to 2 in ruling that Baker's case was justiciable, producing, in addition to the opinion of the Court by Justice William J. Brennan, three concurring opinions and two dissenting opinions.
The majority opinion set up a fairly sophisticated test (which I won't cover here). What is more suggestive is the rejected/dissenting opinion:
Appellants invoke the right to vote and to have their votes counted. But they are permitted to vote and their votes are counted. They go to the polls, they cast their ballots, they send their representatives to the state councils. Their complaint is simply that the representatives are not sufficiently numerous or powerful.
Clearly that view was considered too simplistic by the majority of the court, so simply saying that being allowed to vote (while another's vote counts a lot more e.g. by a corporations-for-whites vote-boosting scheme) is not something that is likely to fly, unless the Supreme Court decides to overrule some prior decisions in this area.
There are more complicated cases in this area, including a recent one Alabama Legislative Black Caucus v. Alabama (2015) decided only 5-4. So, who knows, maybe a corporations-can-vote law could fly in a court with Trump-appointed justices. It might depended on the details of the case.
TL;DR: No, in USA in current case law. Yes, in theory in general.
Corporate person-hood as described in Wikipedia has a two-faced nature: it is both a general concept (a view of a legal system that allows some of laws and rights applicable to persons, to also apply to groups of people acting as a whole through a corporation); as well as a practical body of case law detailing which specific rights and laws apply.
In the United States of America, in current case law, the right to vote has not been granted to corporations.
However, that isn't always the case.
For example, in the City of London elections, corporations vote:
The City has a unique electoral system. Most of its voters are representatives of businesses and other bodies that occupy premises in the City. Its ancient wards have very unequal numbers of voters. In elections, both the businesses based in the City and the residents of the City vote.
Moreover, even in USA, there are some people calling for corporations to have a right to vote:
Montana state lawmaker introduced a bill in 2013 (HB0486) that would grant corporations rights to vote in municipal elections.
... if a firm, partnership, company, or corporation owns real property within the municipality, the president, vice president, secretary, or other designee of the entity is eligible to vote in a municipal election.
tl;dr: Currently no, but it might not be unconstitutional for individual states to grant them voting rights.
Details of voting rights are a state matter. The US constitution has several amendmends which define a minimum of voting rights, namely the 15th (citizens with any skin color can vote), 19th (citizens with any kind of genitalia can vote) and 26th (citizens older than 18 can vote). But all these amendments explicitly grant the right to vote to any citizen, not any person.
While corporate personhood means that a company is considered a legal person in some situations (they can enter contracts, can sue or get sued, they need to pay taxes, they have freedom of speech etc.) corporations are not citizens. The 14th amendment of the US constitution defines citizenship as follows:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
Corporations are founded, not born. So they can not be citizens by birth. Now you could discuss if it would be possible for a legal person to become US citizen through naturalization. I am no US immigration lawyer, and the legal base for the process is long and complex, but I doubt that the process is open to non-biological people.
But while the US constitution provides restrictions for the states about which kinds of voting rights must not be denied, it does not provide any guidance for additional voting rights which must not be granted. So it would theoretically be possible for a state to grant voting rights to corporations. The answer by user4012 mentions that there was a proposal in Montana to grant corporations voting rights in municipal elections. However, I am not aware of any state doing this or planning to do this for any federal election like the presidential election.