Which Christians refuse vaccinations on religious grounds? There are reports of a measles outbreak in a Christian school in British Columbia, Canada. News reports speak of this as a community with "traditionally low vaccination rates" and say that many parents do not vaccinate their children on religious grounds.

I'm unaware of any Christian group that prohibits vaccinations, or any line of theological reasoning that would lead to an issue with vaccinations. Wikipedia doesn't mention anything. Even Amish and Old Order Mennonites appear to go along with vaccinations. 

Who are the groups that take this stand, and what is their theological reasoning?
I believe the answer is to be found in another news article, published around the same time as your question, titled, &quot;Chilliwack [British Columbia] pastor tells congregation vaccines interfere with God’s care&quot;:

The pastor for the community at the centre of the Fraser Valley measles outbreak says he sees vaccines as an interference with God’s providential care.
Rev. Adriaan Geuze says his 1,200-strong Reformed Congregation of North America in Chilliwack mostly shares that view, which is why vaccination rates in the community are “very low.”
“We leave it in (God’s) hands. If it is in his will that somehow we get a contagious disease, like in this case the measles, there are other ways, of course, to avoid this. If (we get sick), he can also heal us from it,” he said in an interview Friday.
...
Geuze [explains] that there is no need to make a healthy “God-given” body “a little bit sick” through vaccination.
He does not oppose other means of boosting immunity, such as rest, healthy living and eating well. Nor does the church oppose medical treatment when a person is already sick, he added.
Asked if he actively advises his congregants not to vaccinate their children, Geuze responded: “Of course I openly express my own point of view according to the Bible, absolutely. But it’s not that we force them. It’s through their own conscience that they have to act,” he said. “They expect that from me, that in a clear way I lay it all before them.”
Opinions on whether or not to vaccinate, however, are divided within the Fraser Valley’s Christian community.
Rev. Abel Pol of Chilliwack’s Canadian Reformed Church said that while he has never surveyed the church’s 400-plus membership on the issue, he suspects that most if not all the congregation is in favour of vaccinations.
...
Pol said those who oppose vaccinations on religious grounds commonly quote the same passage of the Bible, Matthew 9:12, part of which reads:
“And as Jesus reclined at table in the house, behold, many tax collectors and sinners came and were reclining with Jesus and his disciples. And when the Pharisees saw this, they said to his disciples, ‘Why does your teacher eat with tax collectors and sinners?’ But when he heard it, he said, ‘Those who are well have no need of a physician, but those who are sick do. Go and learn what this means: I desire mercy, and not sacrifice.’ For I came not to call the righteous, but sinners.”

So they oppose it based on a reading of Matthew 9:12, a connection I don't understand. The reason they give is that it is &quot;an interference with God's providential care.&quot; Given the church's Dutch reformed background, I suspect it's accurate to pinpoint Heidelberg Catechism Q&amp;A 27 as the source of their discomfort, as the other answer conjectures.
Geuze's church is one of four in the Reformed Congregations in North America, and according to Wikipedia makes up 1251 of 1798 (70%) of the denomination's members. The denomination formed by splitting from the Netherlands Reformed Congregations, another small (though larger) conservative reformed denomination, in 1967. And in fact, the school at the center of the story, Mount Cheam Christian School, is affiliated with the Netherlands Reformed Congregations, and the denomination has its own congregation in Chilliwack.
Two other articles indicate that the NRC prohibits vaccinations, just as its offshoot does, and that there have been previous outbreaks tied to this community's refusal to vaccinate. Those outbreaks have occurred not only in the Chilliwack area, but also elsewhere in Canada.
It's clear that, just as the other answer conjectured, this all traces back to certain conservative reformed folk in the Netherlands:

The anti-vaccination beliefs of conservative streams of the Dutch Reformed tradition remain a concern to public health officials in the Netherlands.
A report in the 2001 American Journal of Epidemiology reported 275,000 members of the tradition in that country (roughly one-eighth of the total) were refusing vaccinations.
The unvaccinated Christians were the only victims of a Dutch polio outbreak in the early 1990s. Virtually all were members of conservative or orthodox branches of the Reformed tradition that make their home in the Netherland's so-called Bible Belt, known as &quot;De Bijbelgordel.&quot;