A local BSA Council is run by the Council's Executive Board. The Board is made up of local businesspeople, community leaders, and usually a few Scouters. Representatives of each organization (churches, VFW Posts, PTA/PTOs, etc.) that sponsors Scout units (Packs, Troops, Crews, Ships) elect the Board members at the Council's annual meeting. These elections are rarely contested; they're usually a slate presented by the previous Board that's elected unanimously. The sponsoring organization representatives are generally not too attentive to this process.
The Council is a not-for-profit corporation; the Executive Board is it's Board of Directors. It's granted a charter (renewable annually) by National Council to operate the BSA program in a particular geographic area. The Council is repsonsible for supporting the sponsoring organizations, signing up new ones, raising money for and supporting Council-wide programs and properties (like the Council's summer camp and headquarters building), etc. To this end the board hires a Council (or Scout) Executive (think Council CEO) and various executive and support staff under him or her. The executives are trained and approved by National, but they are hired and paid by the local Councils.
Units (and their sponsoring organizations) recruit unit leaders. As of April, National conducts a simple background check, to see if you've got any felonies or child-abuse related offenses on your record. But it's the sponsoring organizations who are responsible for determining that a unit leader has the proper character, morality, and patience to work with children, parents, and the unit's other Scouters. The local Council will only get involved if the Scouter becomes publicly infamous for some reason.
A unit and it's sponsor can refuse to register either a Scout or a Scouter for any reason it chooses. A church can require that a Scouter be of it's faith. A VFW Post can decide that it's Troop is getting too big. A PTA can decide that if the parent won't get involved as a volunteer, the kid can't join. Personal hygeine can figure in. A divorced person, or someone living with someone they're not married to, can be judged as being immoral and unfit. A gay or lesbian can only become a unit leader if the sponsor allows it. And if they're not publicly out and make no reference or act to reveal it, especially in a Scouting setting, the Council is unlikely to know, or to do anything about it, given "Don't ask, don't tell".
Regardless of either the morality or fitness of the person who's the unit leader or the tenets of the sponsoring organization, parents would be well advised to get personally involved with their son or daughter's BSA unit. I don't care what you think you know. You never know.
As far as "morally straight" goes, there are numerous sponsoring organizations of units in the BSA that believe that homosexuality per se is not in and of itself immoral. That would include some churches, and numerous secular organizations (who sponsor units containing roughly about half of all Scouts).
What's likely to happen here is that National will ask the Executive Board President (notice that the Scout Executive is nowhere to be heard from; I seriously doubt if he or she asked for this) to explain himself. They'll want to know if this is another variant of "Don't ask, don't tell", or if this means that the Council will accept out homosexuals who will make their orientation and their connection to Scouting public. If that's so, National will look to see if the whole Board agrees with this. If the Board persists, National will threaten to refuse to renew their charter.
What happens next is theoretical, as it hasn't gotten beyond that yet. But they may well solicit another group of concerned citizens, one that will uphold National policy, to apply for a charter and award it to them. Either that, or merge the Council to one that's contiguous to it that is orthodox.