Most of the victims reported being assaulted when they left the camps to get food or wood, MSF said.
In three sites - Rusayo, Bulengo and Kanyaruchinya - more than half the victims said they were assaulted by armed men, MSF said, without giving further details.
Rape has been widely documented as a weapon of war used by armed groups in eastern DRC, where dozens of militias are active. Conflict has simmered and periodically flared up since the end of two regional wars between 1996 and 2003.
"For months, our teams have been treating a high number of cases but it has never reached the catastrophic scale of recent weeks," Jason Rizzo, MSF emergency coordinator in North Kivu province, said.
"Our data only reflects people who arrive at our facilities so it's probably underestimated," he told Reuters.
A staff member from another humanitarian organization, speaking on condition of anonymity, said concerns had been raised with DRC authorities about alleged rapes being committed by members of the military and allied proxy groups, and about reports of displaced people being prostituted in brothels run by army officers and local authorities.
Lieutenant General Constant Ndima, military governor of North Kivu province, said he could "not rule out" that members of the military had committed sex crimes.
"The victims speak of armed men. As you know, armed groups are currently scattered around Goma," he told Reuters, adding that investigations were under way.
"Let's wait for the conclusion of the experts' work," said Colonel Guillaume Ndjike, army spokesperson for North Kivu province, when asked to comment on the allegations.
North Kivu and neighboring Ituri province have been governed by the military for the last two years since the DRC declared a state of siege to address worsening violence.
Most of the recent displacement in North Kivu province has been caused by fighting between the DRC army and the M23, an ethnic Tutsi-led rebel group.