While the CDC can collect and publish data, they cannot push an agenda. The problem lies in the fact that the unbiased data doesn't support their gun-control agenda.

Via the National Review: 

One of the common talking points that liberals throw around in the gun debate is that Republicans have banned even studying gun violence. So you get headlines such as “Lift the Federal Ban on Gun Violence Research” (the New Republic), or “Why Gun Violence Research Has Been Shut Down for 20 Years” (the Washington Post’s Wonkblog), or “GOP Chairman: Congress Should Rethink CDC Ban on Gun Violence Research” (The Hill), or “What’s Missing from the Gun Debate. It’s Simple: Science” (an op-ed in Politico).The reality is different, and it illustrates two contending views of how America should be governed.

To start with, nobody has been banned from anything, in the way we typically think of government bans...

So, why are gun-control advocates up in arms (rhetorically) about the CDC? Cue the 1996 Dickey Amendment, regularly renewed by Congress, even after its author, Jay Dickey, left Congress and changed his tune on the issue, according to his 2015 interview with NPR’s Steve Inskeep. This amendment was attached to the funding of the CDC and provided a restriction: “None of the funds made available for injury prevention and control at the Centers for Disease Control and Prevention may be used to advocate or promote gun control.” But the CDC does still collect its own empirical data, published regularly as part of its National Violent Death Reporting System...

As Rosenberg himself wrote in that Politico op-ed, “What’s Missing from the Gun Debate,” the result of the Dickey Amendment was to roll back his use of the CDC for that purpose:

The amendment did not explicitly prohibit the CDC from conducting gun violence research; it prohibited the CDC (and later, other federal agencies like the National Institutes of Health) from lobbying for gun control legislation. Nevertheless, the provision was a shot across the bow and had a chilling effect. A second shot was Congress’ taking away the $2.6 million that the CDC’s injury center had been spending annually to support gun violence research. The third shot was fired by CDC itself, when the agency director fired the person most closely identified with the gun violence prevention research. (That person was me.)...

So, why shouldn’t the CDC get taxpayer money to do these studies? There are two main reasons, one relating to the nature of social science in general and public health in particular — and the other relating to the nature of democratic self-government...

Naturally, if you hand over funding to people with an agenda, and you place no restrictions on its use to promote that agenda, and their work is peer-reviewed (if at all) only by people supportive of that agenda, and the underlying scientific discipline has no real methodological guardrails, you will get research not only that is biased but that gets progressively less rigorous the less supervision it can expect...

By all means, let’s keep collecting data and debating its meaning. The gun debate in America will never be over; nor should it be. Even the Bill of Rights can be changed if the people are determined that some rights just aren’t worth protecting anymore. Defenders of gun ownership should not fear that debate. But we shouldn’t be delegating such important social questions to agenda-driven advocates operating behind the illusion that they are doing disinterested scientific research. We shouldn’t let them spend our money to support only one side of a debate over a right enshrined explicitly in the Constitution.