Approximately 250,000 people have joined the organization's existing 4.25 million members, according to NRA spokesman Andrew Arulanandam.
In addition, the NRA is receiving an influx of financial contributions, he said, adding: "This is going to be a very expensive and hard-fought fight."
A CNN/Time Magazine/ORC International poll Wednesday indicated that Americans generally favor stricter gun control, but they don't believe that stricter gun laws alone would reduce gun violence.
According to the survey, 55% of Americans generally favor stricter gun control laws, with 56% saying that it's currently too easy to buy guns in this country. However, only 39% say that stricter gun controls would reduce gun violence all by themselves.
Obama called for citizens to let their elected representatives know what they think, saying: "The only way we can change is if the American people demand it."
He proposed legislative steps he previously has backed, such as reinstating the Clinton-era ban on assault weapons, and also requested that funds be made available to help treat mental illness and provide schools with support to enhance their safety.
His executive actions signed Wednesday called for tougher enforcement of existing laws and required federal agencies to provide data for background checks.
A senior administration official told reporters the price tag for the entire package was $500 million.
Obama also said he would nominate B. Todd Jones, the acting director of the Bureau of Alcohol, Tobacco, Firearms and Explosives, to become its permanent chief. The agency has been without a full-time director for six years.
A main focus of Obama's steps was closing loopholes in background checks. While requiring universal background checks would require congressional approval, some of the executive actions signed by Obama were intended to bolster the existing system.
Across the country, more than a million people failed background checks to buy guns during the past 14 years because of criminal records, drug use or mental health issues, according to FBI figures.
That figure, however, is a small fraction of overall gun sales.
"If you look at the combination of likelihood of passage and effectiveness of curbing gun crime, universal background checks is at the sweet spot," said Sen. Chuck Schumer, D-New York, a leading backer of such screening.
Obama also called for more money to strengthen gun safety at schools, including hiring more counselors such as retired law enforcement officers to help educate students on gun issues. He also called for more funding for communities to hire more police officers, but stopped short of seeking the NRA's proposal for armed guards at every school.
The CNN/Time/ORC poll Wednesday showed that respondents favored armed guards in schools by 54%-45%.
Congressional hearings planned
Legislators said working with Congress will be paramount in curbing gun violence. California Rep. Mike Thompson told CNN on Tuesday that a ban on high-capacity magazines could garner Republican support, but a full-scale assault weapon ban would be hard to get passed in the GOP-controlled House.
House and Senate committees said they would start holding hearings on gun control measures in coming weeks.
In New York, Democratic Gov. Andrew Cuomo on Tuesday signed into law a series of new gun regulations -- the nation's first since the Newtown shootings.
Both New York's GOP-controlled Senate and Democrat-controlled Assembly approved the measure by overwhelming margins.
It includes a statewide gun registry and adds a uniform licensing standard across the state, altering the current system, in which each county or municipality sets a standard.
Residents are also restricted to purchasing ammunition magazines that carry seven bullets, rather than 10.
Keene derided outlawing high-capacity magazines as "a bidding match" that focuses on the wrong issue.
"So the president says you don't need 30-round magazines. How about a 10-round magazine? Andrew Cuomo says, 'Well, I can do better than that. I'll make it a seven-round magazine,'" Keene said.
"The fact of the matter is the kinds of people who do this, particularly the mentally unbalanced -- who are the most likely people to do it -- shouldn't have any magazines," he said.