Not necessarily. You can design a nationally-representative survey with few respondents as long as your survey is designed well, and you achieve a strong response rate. You can actually check the math here: Sample size calculator - CheckMarket
For example, I entered 300,000,000 for the population size (just to get a nice round number. past a certain point, population size doesn't really mean much), a response rate of 35% (the only thing that a higher response rate achieves is increasing the number of people invited to the survey), with an MoE of 2% and 99% confidence interval. That resulted in about 12,000 people people being surveyed, and a sample size of 4147 (again, response rate 35%).
It is, however, a little difficult to tell exactly how they structured the survey and what the response rate was because it looks like they haven't published the study yet. I'd be interested to see what their methodology was. I'm also curious as to some of the claims in the article. For instance, it said that the survey sampled approximately 4,000 gun owners, but it also stated that, "The study found that 22 percent of American adults say they personally own a firearm. " I'm not sure whether there was a distinction made between having a firearm and OWNING a firearm (e.g. having a gun in the home vs. personally owning one), or if that's an oversight by the writer. Regardless, if 4,000 gun owners were included in the sample, I don't see how you could draw out the 22% ownership rate from that. I can definitely understand the line about 3% of gun owners having half of the guns, but I don't get the 22% figure.
Again, I'd like to see the survey itself before drawing conclusions about the sampling method, but the statistical reasoning has the potential to be sound.
Can you repeat that for those of us on crack?