I haven't seen any non-religious Republicans. Is it a present qualification for politicians running on the GOP ticket to be religious?
First, there are atheist republicans, to answer your title. Not a lot, but there are.
Elected politicians or political operatives
Charles T. Beaird (1922–2006): Republican Party member/politician and newspaper publisher.
Clint Eastwood, according to his biography.
Robert Ingersoll (1833-1899) was a prominent Republican politician.
Karl Rove, a major Republican strategist responsible for G.W. Bush's election.
Ayaan Hirsi Ali is an interesting case. She's a political figure in USA (with American Enterprise Institute, a conservative think tank); and a politician - but she's an actual right-wing politician in Netherlands, not USA!
S.E. Cupp is a pretty prominent conservative figure.
Edwina Rogers was a conservative lobbyist with a high profile, and a former Executive Director of Secular Coalition for America (ironically, speculation is that she was fired for being a Republican though that was never proven).
If you're willing to include people who aren't strict atheists, but closer to agnostics or what fits into now-trendy term "nones", you have such conservative notables as Charles Krauthammer and George Will.
Honorable mention: Ayn Rand.
While militantly atheistic Ayn Rand was openly antagonistic to conservatives of her time (who - according to her - were openly pro-religion and insufficiently pro-capitalist), she was even more antagonistic to progressives on most main political/governance issues.
And at this point in time, while she is nearly universally reviled by Democrats/progressives; in an ironic twist, both her and her Objectivist philosophy is a heavy influence (and wildly respected by) many Republicans and conservatives today, ever since Reagan. One of the reasons for this twist is the late tactical convergence of libertarians with Republicans and Conservatives). John Galt was far more of a rallying cry for Tea Party than Jesus was, especially at its inception).
Second, there is possibility that some Republican politicians are atheist but aren't vocal about it.
This makes sense both for:
tactical political reasons (progressive atheists aren't likely to vote for a Republican even if he's an atheist so no votes to be picked up there; a far smaller # of people who are conservative atheists similarly wouldn't require politician to be atheist to vote for him in the first place; and non-atheists wouldn't be more likely to vote for an atheist over his professed lack of belief and may be more likely to NOT vote for the person if they are deeply religious).
Mindset. A person who's a conservative/Republican politically and atheist faith wise, typically self-identifies far more as the former than the latter, and as such would be less inclined to discuss their religious views in the first place in public (even witout concern of their political ambitions).
Please note that there are at least 9 members of 114th Congress who didn't disclose their religious affiliation. Unfortunately, Pew didn't break down the party lines, so it's unknown how many of them were D vs R.
Now, let's examine possible reasons why:
First of all, it's not really a "Republican"-only phenomenon. It's a general fact of American politics.
There are precious few atheist Democrats either (Pete Stark is the only federal level atheist politician I can name off-hand, and he's not technically a self-admitted atheist, he's a Unitarian). The wiki page linked above has only 5 other examples of national level Democrat politicians, past or present, who are atheistic; plus the ever-nonconforming Jesse Ventura who's an Independent.
This is influenced both by:
Dearth of atheists in USA. Depending on which poll and datapoint and question formulation you take (ex1, ex2, ex3), pure atheists (as opposed to agnostics or no-declared-religion people) in USA make up between 1% and 3% of population, and even far wider "no declared belief" category i only ~10%.
Of course, this makes the potential pool of atheist candidates pretty thin to begin with.
53% of people being disinclined to vote for atheists in office (and only 5% more inclined - src).
Republicans specifically have a large contingent of highly-religious voters.
This came about partially as part of "Movement Conservatism" in the 1980s.
That resulted in the fact that a lot of the most active voters and base members consider religious faith to be important, in general.
There is a perception (the fact whether it's right or wrong is irrelevant - just the fact that perception exists and is common) that atheism is incompatible with having morals.
This perception has two components:
Theoretical ("if someone has no fear of god or being judged in the afterlife, they have no incentive to be moral")
Empirical ("Looking at examples of openly atheist countries around, we see awful USSR with all its atrocities, Red China with all the wonders Mao did to his people, and North Korea").
This perception is stronger among conservatives/Republicans.
The rise of libertarianism.
People who are conservative but atheist tend to lean to fiscal conservatism and social non-conservatism. Lately, they basically drift towards libertarians.
And the rise of libertarianism (small "l") as a political force gave such people an outlet both for running for office and for voting as Libertarians (big "L") instead of Republicans, when they find that Republican party became too attached to Religious Right's social conservatism agenda.
As one famous example, Jesse Ventura openly supported Libertarian candidate in 2012 after Romney won the nomination for Republican presidency.
I can't find it right now but I saw polls that showed a potential of a ton of libertarian support for Ron Paul if he ran independently of "R" ticket. While not an atheist, he clearly fits the political mold (fiscal conservatism, largely-non-religious-based social views).
As per Pew,
About two-thirds of atheists (69%) identify as Democrats (or lean in that direction), and a majority (56%) call themselves political liberals (compared with just one-in-ten who say they are conservatives).
Historically, atheism was dominated by progressives/left wing; and they mounted an extremely effective campaign to associate Republicans/conservatives/"right wing" with anti-science/anti-intellectuallism and as incompatible with atheism.
How do we know that's indoctrination and not a feature of atheism? Because we have a control group. People who grew up outside the influence of western colleges but in atheist environment form a large and well-polled demographics - immigrants from former USSR. They are heavily non-religious, and nearly overwhelmingly vote for Republicans.
I could not find a living one or one who holds high office, but I found Charles T. Beaird, a Louisianna Republican in the 1950s.
From his obituary, Charles T. Beaird "was a 'nontheist,' meaning in simplistic terms that the concept of God was not among the ideas on which he based his beliefs — but that if you did, it was fine with him."
From Wikipedia "In 1956, he was elected to the Caddo Parish Police Jury, the equivalent of county commission in most other states. He was one of the first Republicans elected to public office in Louisiana since Reconstruction. ... Beaird attracted national attention in 1956, when he gave a seconding speech for the renomination of President Dwight D. Eisenhower at the Republican National Convention at the Cow Palace in San Francisco, California. "
So Beaird was an atheist (or at least agnostic) and a Republican politician.