Does Spider-Man meet the requirements to register?
According to the New York Board of Elections site, here is the criteria to register to vote:
Be a United States citizen
The application form simply asks you to check Yes or No and does not require any accompanying documentation. So since Peter Parker is a United States citizen, he can truthfully check this off on his Spider-Man voter registration.
Note that there was a 2014 bill to allow illegal immigrants in New York to be allowed to vote. Should a bill like that pass, that would probably waive this requirement.
Be 18 years old by December 31 of the year in which you register
Spider-Man hasn't always been over 18 (he was in his mid teens in his early appearances), but we'll assume that he is currently eligible. Like with his citizenship, no documentation is necessary to register.
Note that when Spider-Man debuted in 1962, the minimum voting age was 21. The Twenty-sixth Amendment was ratified in 1971 reducing the age to 18.
Live at your present address at least 30 days before an election
I believe that Spider-Man could pass this requirement without providing a documented address because homeless people are allowed to vote in New York. From the National Coalition for the Homeless (emphasis mine):
Pitts v. Black, 608 F.Supp. 696 (S.D.N.Y. 1984).
Plaintiffs challenged a New York State Election Law provision forbidding people living on the streets from registering to vote. The District Court held that the New York City Board of Election’s application of the residency requirement disenfranchised an entire group of people, which is forbidden by the Equal Protection Clause. The court found that a person’s “residence” is the place at the center of the individual’s life and the place where he/she presently intends to remain. The court reasoned that people need only have a specific location that they consider their “home base” — the place where one returns regularly, manifests an intent to remain, and can receive messages and be contacted.
Spider-Man could consider New York City his "home base", since he returns to it regularly, manifests an intent to remain, and can receive messages and be contacted (which he often does by various signals or a notice in the Daily Bugle).
Not be in prison or on parole for a felony conviction
Despite all the crimes that he's been accused of, Spider-Man has been fortunate enough to avoid prison and parole, so he has no problem with this.
Not be adjudged mentally incompetent by a court
J. Jonah Jameson may call him mentally incompetent, but a court never has.
Not claim the right to vote elsewhere
I suppose that if Peter Parker claimed the right to vote, then this would legally prevent Spider-Man from registering to vote, since they are one and the same. But if Peter Parker did not claim the right to vote, I think that he would legally be in the clear on this.
So could he actually vote on Election Day?
First time voters in New York are required to provide identification on Election Day. From the online registration form (emphasis was in the original):
We’ll try to check your identity before Election Day, through the DMV number (driver’s license number or non-driver ID number), or the last four digits of your social security number, which you’ll fill in below.
If you do not have a DMV or social security number, you may use a valid photo ID, a current utility bill, bank statement, paycheck, government check or some other government document that shows your name and address. You may include a copy of one of those types of ID with this form— be sure to tape the sides of the form closed.
If we are unable to verify your identity before Election Day, you will be asked for ID when you vote for the first time.
Spider-Man doesn't have a driver's license (despite the fact that in one incarnation he had the Spider-Cycle) or a social security number, so he can't use that. Since Spider-Man doesn't have utilities or a bank account under his name. However, he has gotten paychecks before, like in his first appearance in Amazing Fantasy #15 where he worked as an entertainer:
Given that he has trouble cashing paychecks, there would probably be some who would probably doubt whether or not that was valid.
However, there is the option of a government document with a name and address. Presumably S.H.I.E.L.D., being the branch of a government that deals with superheroes, could issue such a document. If nothing else, some sort of government dossier that says "Spider-Man, works in New York City" would meet the letter of the law.
With either a paycheck or a government document, Spider-Man would be legally allowed to vote.
Will he have to prove it's him on Election Day?
One issue that Spider-Man might face is same one he faced when trying to cash a check in his first issue: how can he prove he's Spider-Man and not someone else dressed in the same outfit? Given the number of supervillains who have tried to impersonate him over the years, it's not an unrealistic concern. Probably he could get another superhero (Tony Stark?) to sign an affidavit saying that it is him.
Can Spider-Man vote if he signs an affadavit?
As of 2012, New York allows voters to cast ballots by affadavit saying that they are legally registered to vote. This was due to Hurricane Sandy destroying many polling sites, so voters were allowed to go elsewhere to vote. Since polling sites only have records for their own voters, a signed affadavit was permitted.
Given all this my answer is that Yes, Spider-Man can register to vote and would be legally allowed to actually vote on Election Day. However, he's going to have a lot of public opinion resistance from J. Jonah Jameson (and renewed calls for Voter ID laws in future elections) if he does so.