My group runs on similar principles but doesn't have that problem. I'll describe some of the things that we do which might impact who GMs.
Half way through each game, people willing to GM nominate the games they are willing to run. The following week a vote is taken to determine which ones.
- Each game runs for 6 weeks. This is less intimidating than 6 months for new GMs and allows us to try out a lot of different games. The cost of this is that it is harder to develop long term campaigns. (Not impossible though, some games do return time and time again, to the extent that the Exalted game is on the verge of collapsing under the weight of its backstory and the way that second edition mechanics start to fail at high XP levels).
- GMs are not allowed to run more than 3 games in a row (nor are they allowed to run the same game more than twice in a row). This is designed to avoid cliques forming, but in a group where few people are volunteering to GM it could provoke a "Well if I do it, then at least there will be a game" response, unfortunately that has a risk of backfiring.
- GMs of the current games are rewarded with first pick of available player slots for the next block of games.
Of course, for any of this to work, the members have to buy in to the system — and if most of them don't want to GM then that will be a hard sell.
So, that is what we do… but those are approaches at solving speculative problems. You really need to be addressing the barriers that are stopping people who aren't volunteering, and you need to find out what those barriers are before you can do that.
Some more possible solutions for speculative problems:
There are too many rules to learn
Introduce a simpler system to the club and encourage people to run that.
I don't have time to prepare
Introduce simpler systems that avoid the need to spend a long time writing stats for characters.
I have a hard time reacting to what players will come up with
Encourage the use of systems with rules that hand narrative control over to the players (or add house rules). (e.g. Fate or Hillfolk/DramaSystem)
Do I know anyone who could get me a meeting with the mob boss?
(After optional Contacting role)
You do. Tell me his name, job and favourite bar.
As well as reducing the burdon new new GMs, when an experience GM uses rules like these then it will get players used to contributing these details so things will be easier for them when they GM.
Our club Chairwoman hasn’t gotten out of the GM seat for 2 years now. I see this as incredibly unfair.
One of our members (who enjoys GMing far more than PCing) grumbles about how infrequently he gets to GM. He was surprised to discover that nobody GMed more often then he did last year. It is possible that your chairwoman prefers sitting behind the screen, be careful you aren't trying to solve a problem that doesn't exist.