Dogs and cats have been invited to vote in the swing state of Virginia, and Mitt Romney smells a rat.

The Republican’s presidential campaign is calling on state officials to launch a criminal investigation into voter registration forms that a District-based nonprofit recently mailed to hundreds of dead Virginians, children, non-citizens, pets and others ineligible to vote.

The Voter Participation Center, which tries to get “historically underrepresented groups” such as women and minorities to vote, acknowledged that it had addressed some forms to people and animals with no business going to the polls. The State Board of Elections said “hundreds if not thousands” of applications were sent to ineligible voters.

The center blamed a faulty commercial mailing list and said the errant mailings represent a small fraction of the nearly 200,000 forms it has sent out across the state.

The mailings have revived talk of voter fraud in Virginia, a crucial swing state where President Obama and Romney, a former Massachusetts governor, are deadlocked in a recent poll.

“This presents a very significant risk to the proper administration of the upcoming general election,” Kathryn Biber, the campaign’s general counsel, wrote in a letter sent Tuesday to Attorney General Ken Cuccinelli II (R) and the State Board of Elections, calling for a criminal investigation.

A spokesman for Cuccinelli said Wednesday that any investigation by the office would have to be initiated by a request from the elections board. The board, while sharply critical of the mailings, does not plan to seek an investigation “at this time,” said Chairman Charles E. Judd.

The board has received more than 750 complaints, mostly from people whose pets or deceased relatives have received solicitations to register to vote, Judd said.

The center initially responded to complaints with a comical video that shows a dog talking about how unlikely it is that a cat could vote.

“Someone sent the cat a voter registration form,” the dog says. “Cool out, cat. If you even tried to vote, election officials would catch you. You’re not eligible, you’re not 18, you walk on four feet and you cough up hairballs, which is gross.”

Elections officials were not amused, chastising the center for its “flippant” video in a letter that also said the group violated state law by filling out applications with voters’ names. The law requires that the form be filled out by the applicant or by dictation.

Officials at the center have since dropped the lighthearted tone and have stressed that they mailed applications for registering to vote — forms widely available at government offices and online — and not voter ID cards, which can serve as identification at the polls and can be issued only by elections officials.

“We have nothing to do with that issue, voter fraud. We send people applications to fill out in the mail,” said Page S. Gardner, the center’s president. “It’s up to them to fill out the form and obey all the state laws and federal laws.”

But for some, the mailings have reignited fears that Virginia is vulnerable to voter fraud, an issue that has been bitterly debated around the country in recent years. Citing concerns about the integrity of elections, Virginia’s GOP-controlled General Assembly this year closed a loophole that had allowed voters to cast ballots without showing identification. Democrats charged that the voter ID law, while more moderate than versions Republicans have recently pushed in other states, was intended to make it harder for minorities and other Democratic-leaning groups to vote.

State Sen. Thomas A. Garrett Jr. (R-Louisa), a Louisa County prosecutor who successfully tried two felons who registered to vote in 2009, said one of the two registered with a form mailed to her by the Voter Participation Center.

“Clearly they haven’t gotten the message,” Garrett said.

In her letter, Biber contends that the center’s mass mailing may have violated state laws, including those that prohibit falsifying a registration application and communicating false information to voters about their registration status.

Gardner said the group tries to make its mailing list “as perfect as possible.” Dead people can wind up on a mailing list because it is compiled from things such as magazine subscriptions, which often are not updated with a new name when a spouse dies. Some people have subscriptions in the names of their pets for reasons that Gardner, who described herself as “a non-pet owner,” said she did not understand.

“This is very sloppy prospect mailing going on,” Judd said. He said he was particularly concerned that the mailings told recipients that “records show that you are eligible to vote in the 2012 presidential election.” That wording makes it sound as if the center is an official government elections entity, he said.

The focus on the errant mailings is a “man-bites-dog story” in Gardner’s view, one that she said misses the bigger picture — that nearly 2 million eligible Virginians are not registered to vote.

“It’s fun to write about Mozart and other pets getting these voter registration applications,” Gardner said, referring to a dead dog who was sent a form from her group. “[But] at some point, we have to look at ourselves and say, ‘Really, what’s the story here?’  ”