Mr. Carson is such a newcomer to Republican politics that he hasn’t voted in a GOP primary in decades, having registered as an independent and with a third party. But his team is setting high goals for 2016. The campaign’s chief executive, Terry Giles, a Houston attorney who is also a newcomer to political campaigns, said he aims to raise $100 million to $150 million for the first four primary states—an unlikely figure for a candidate who doesn’t have a network of big donors and who aides say dislikes making fundraising calls himself.

Mr. Giles said the campaign aims to carry 55% of the popular vote in November 2016. “You’ve got to be double digits for it to be a mandate,” he said…

When he retired to West Palm Beach, Fla., in 2013, county records show Mr. Carson registered to vote as a member of the Independence Party of Florida, a relatively obscure centrist group with 49,000 members.

Peter Allen, a suburban Tampa electrician who founded the party in 1999 and ran as its candidate for governor in 2010, described its politics as centrist. He said he is for tax cuts but also praised the health law. “I think Obamacare is one of the best things to come out of the White House in 35 years,” he said.

Mr. Carson was unaware he had aligned himself with the minor party. “This is news to me,” he said. “As far as I know, I was registered as an independent.”