I have done background check on my employees and unless they do it in other states, there is no mention if you are a CCW or not. I have a service that does a basic criminal Background check (Federal, state, local, and Interpol) and credit history and score. The check usually comes back with any crime that the person commited, with a where and when. The 2nd half is your typical credit history (who you owe money to and how much) with a credit score, it also verifies last know address. Some people might get offended about checking on credit scores, but people who are doing really bad money wise, could be more tempted to steal or commit fraud. I usually do a Google search (never leads to anything good, but you never know), and call the last couple employers (I don't call current employers, I don't want to get the applicant in trouble with their current boss), to verify employment. That is about it.
I can't find out if you have a permit to carry, a Fishing license, or an AARP card. You cannot get the info through regular sources. Try calling the whatever entity in your state that provides CCW, and ask them if you can find out if your neighbor has a permit and see what happens. The only way they can get access is if they are a government agency, and if your getting some sort of high end background check for a position in a government agency, that position would probably be with Law Enforcement or the Military and I bet they either will not care or will be happy that you already have some prior experience with firearms.
With helping my cousin last year fill out job applications for some State and federal Law enforcement positions (Border, Secret Service, FBI) they do want to know how many weapons you own, what the make, model and serial # is and if you have a CCW. He had several guns and a CCW (he graduated with a Criminal Justice degree, and always wanted to be a cop). He had no problems regarding the weapons or the CCW. He chose to go with the Secret Service and now handles counterfeiting & large scale credit card fraud in NYC. When he went for his polygraph they asked him questions about his guns and his CCW, basically seeing if he owned any other guns not on the list, and to see if he ever carried without a permit.