Q:

How to encourage new GMs (in a club environment)

I am part of a games club that has been running for years. Our size varies but we usually have around 10-15 members and run 2-3 games at a time. We change games every 6 months (this also varies sometimes).
Only about half of those members actually GM, and most of those only do this occasionally (and often grudgingly).  Our club Chairwoman hasn’t gotten out of the GM seat for 2 years now. I see this as incredibly unfair.
I would like some advice on how to "encourage" players who have never GMed before to take the plunge. 

A:

The root question here is Why are some members not keen on GMing?  This could be answered by a simple questionnaire (if you had the time to do it) or by asking a sample of them face to face. Now, because you are not asking about that, I'll leave it at that for this side. 
There are tricks that you can do:

Have a library of ready to run adventures.  Once a set of players have played it, one of them runs it for a new set of players. 
Bribes.  Things like pizza, beer, wine, or just a lift to where the game is played. Maybe even a T-shirt with $SuperCoolImage as a prise for being a first time GM. 
Positive feedback. Make sure that the new GMs can talk to the old hands and get help. Hey, this site might be a good source of help too... 
Encourage players to come up with settings elements while they are playing. This will get them into a designer mode which is what most GM do in preparation for a game.
Limit GM time: If you GM a game, you must play in one next time.  Thus, some players will have to step up to get a new game running. 
Make sure the new GMs run games in settings they are more familiar with than the players.
New GM could run shorter games -- maybe even one shots -- to get more familiarity with the science of GMing. 
Pair GMing.  Have two GMs, one experienced and the other not to GM a game at the same time. It will require a little setting up but the rewards are great. This has nothig to do with pair programming, you know...

Note that some players do not want to GM at all.  No amount of manipulation will get you to make them run a game. This should be fine too. 

A:

The club I used to go to had a (small) entry fee and a raffle at the end, to help with the cost of renting the place, mainly. GMs didn't have to pay for the entry and they got a free raffle.

A:

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.

