Does the NSF support researchers with no academic affiliation? It seems that NSF only supports researchers affiliated with academic institutions. For instance, if a researcher is in the industry, is there a chance to get funding from NSF for a pure academic research? 

Let's say the researcher's employer is not a scientific/research establishment, maybe it's a manufacturing plant, and the researcher's interest has nothing to do with his/her daily job.
Another answer quotes the Grant Proposal Guide, which is of course the essential resource.

The NSF does solicit proposals (like other agencies) for the Small Business Innovation Research (SBIR) program.

But I'd also say that whenever you have questions about NSF funding for a research idea, I'd figure out an appropriate program officer and send them an e-mail. For example, I'm in chemistry, so I go to the NSF chemistry website and guess an appropriate sub-program.

In general, I send an e-mail saying something like:


  Hi, I have a research idea that I think might fit in your program but I have some questions. Would there be a good time to talk this week? [etc.]


Sometimes I'll even get a call back promptly from the program officer, or a polite message that program X might be a better fit, etc.

In all cases, the NSF program officers want to fund the best research they can, so they are very helpful at suggesting appropriate approaches (e.g., if it's possible for them to fund your idea).