One of the changes that the GSUSA went though over the last decade was to allow youth to remove the word "God" from the GSUSA oath. This was justified by the GSUSA to WAGGGS (World Association of Girl Guide and Girl Scouts, the international GS sanctioning body) on the basis that the GSUSA wanted to allow girls for whom "God" was too Judeo-Christian to substitute their own deity (a.k.a Allah, Jehovah, Brahma, etc.). WAGGGS requires that it's membership organizations require their individual members to profess some kind of higher power. However, the effect seems to have been to allow the GSUSA to accept members who substitute nothing for "God", either verbally or mentally. And it seems that the GSUSA is doing little to disabuse people of that here in the U.S.
As far as your son goes, if he's 7 he's Tiger Cub or Wolf Cub age, depending on his grade level (1st or 2nd grade; if he's homeschooled, you can pretty much pick it yourself). Look in the phone book or yellow pages for the name/address/phone number of your local Council. Or, go on here, enter in your zip code, and get it from there.
Call the council. Tell them what town you live in, and ask for your District Executive. Once you're talking to him or her, ask about the Cub Scout Packs in your area. Find out who their sponsors are. The DE will want to know what church you go to and what school your son goes to so that they'll know whether or not there's a Pack sponsored by them. By the way, it's possible for you to live in one District, but for your son's school or church to be in another, so you might end up talking to another DE. Get the names and phone numbers of the sponsors and the Cubmasters of 2 or 3 Packs.
Call them up. Ask them how big the unit is, when/where it meets, and what activities they do. Outings, campouts, service projects, etc. A Pack meets once a month, but your son will be most closely associated with a Den, made up of 4 to 8 Cubs his own age/grade level. Find out what Dens they have and who the Den Leaders are, when/where they meet (often in the Den Leader's home), and what the Den Leader's phone number is. Now call the Den Leader and ask the same questions.
Then take your son to a couple of Den meetings, and see if you like what you see. Please understand that a Den Meeting is not usually all that quiet. These are young boys, and while sometimes they're working on a project, a smart Den Leader sets aside some time for the boys to be active and let off steam. In short, to be boys.
Should you find a Den and Pack to your liking, you'll fill out a short application, and your son is a Cub Scout. Congratulations! Expect to be solicited to become a leader. I encourage you to do so. You'll find it rewarding (I have). The application asks for your SSN. They use this to do a background check on you. The BSA NEVER releases information of any kind about their leaders to anyone. Companies have offered fortunes to the BSA for it's mailing list, but it never releases or sells the information and never will. You'll also be asked to take a couple of basic training courses. Please try to fit those into your schedule. You'll find it very helpful (I'm on our Council's training staff).
But, even if you can't sign up as a leader, please consider helping out at either Den or Pack meetings, the popcorn sale, etc. Understand that there's a group called the Pack Committee that handles running the fund raisers, recruiting leaders, buying the awards, getting the newsletter out, etc., that you can do on your own time without having to go to Pack or Den meetings. You can join it.
Of course, if you really want to go the whole hog, you can organize a Pack at your church or other community organization. You only need 5 Scouts and 5 adults to start a Pack, and small units are much easier to handle than some of those Cub Scout Packs with 100 or more Cubs in them.
As to the issue at hand, I don't particularly like the current method of providing information for a background check but can't see a real way to provide the desired level of security against pedophiles any other way. I guess I really prefer to have parents providing the leadership under the Safe Scouting/Youth Protection guidelines. Any outsider should be carefully scrutinized before being allowed to work with the youth.
Again, thanks for your Scouting service and well-supported information in this thread.