I've been climbing for a few years now and have never had to wash my chalk bag. What happened to yours? Did something spill on it? I can't imaging why one would need to be washed unless you are trying to sell it or it got into an accident and got spilled on!
Either way, I don't see why it would hurt to wash it by hand and let it dry out in the sun. I know that some chalk bags have different materials in them besides the fabric, so they may not all be good for the washing machine, but by hand I think is alright, but I have never had to do it before so I am not talking from experience!
I would add that it also depends on if it's indoor climbing or outdoor climbing. I don't climb indoors as much as I do outdoors, but usually when we go out we easily have full day sessions, leaving in the morning and getting back at night, sometimes the next morning even.
I doubt I would be able to have such long sessions in an indoor gym though. There is something about the outdoors that just makes it possible to have such long sessions. I think that for indoors my longest session has been about 3 hours.
Hope this helps!