But Keene said his meeting last week with the vice president left him "disappointed," with little perceived ground for compromise.
"We will not allow law-abiding gun owners to be blamed for the acts of criminals and madmen," the NRA said in a statement after the meeting.
The group is preparing a new ad campaign to fend off certain gun regulations.
Governors in states like Connecticut, New York and Colorado have called for tighter restrictions and extensive background checks during firearms sales. Other states like Alaska, Arizona and Montana have sought to pass legislation that would exempt them from stricter federal regulations.
"The gun issue will be prominent in a lot of states," said Bill Pound, executive director of the National Conference of State Legislatures. "Whether there will be action taken is another question. If experience is any guide, there will be far more bills introduced than enacted."
The city of Burlington, Vermont, sought to get tougher on the kind of weapon used in the Sandy Hook shootings, passing a resolution that could lead to a ban on certain rifles and high-capacity magazines.
Sandy Hook Promise
In Newtown, a group of residents plan to mark the month that has passed by unveiling a grass-roots campaign to prevent similar tragedies.
The group calls itself the Sandy Hook Promise, formerly Newtown United. It aims to "identify and implement holistic, common sense solutions that will make our community and our country safer from similar acts of violence through education, outreach and grass-roots discussion," according to its website.
"It is unfortunate that we need tragedies to address problems," said Brookings Institution fellow John Hudak, who grew up roughly 10 miles from Sandy Hook. "It has been true across the world in responses to terrorism, in the improvement of disease prevention, in the testing of pharmaceuticals and even in the growth and development of democracy."
Regulations and legislation
After Newtown, several congressional lawmakers promised to introduce tougher gun-control measures. On the first day of the new session, lawmakers in the House of Representatives introduced nearly a dozen bills related to gun violence.
Rep. Carolyn McCarthy, D-N.Y., has sponsored legislation that would require background checks for all gun sales -- including at gun shows -- and ban online sales of ammunition.
In the Senate, one such bill is the Fix Gun Checks Act, which attempts to address the "gun-show loophole" and requires criminal background checks for all firearms sales, rather than just licensed dealers.
Sen. Richard Blumenthal, D-Conn., announced that he planned to introduce legislation requiring background checks to purchase ammunition.
"Ammunition is now the black hole in gun violence prevention," he said, pointing to laws on bullet sales that permit exchanges without background reviews.
Some are pushing for a new ban on "assault weapons" to include other types of military-style weapons and those with high-capacity magazines.
The NRA opposes such a ban, saying it won't help and infringes on Second Amendment rights.
In Connecticut, Gov. Dannel Malloy last week bluntly rejected the notion, espoused by the NRA, of putting more guns in classrooms as protection.
All states except Illinois allow at least some limited ability to carry a concealed weapon in public, often requiring "good cause" by an individual before a permit is issued.