I’ve been using a sling at LT events for a year (and stumbled across this thread while googling to see if anyone else was as daft as me). It takes practice, but it is, just about, a viable weapon.
I tried out a variety of sling lengths and ammo, and my current recommendation would be a relatively short one of about 34’’. That sounds a lot, but when folded in half around a rock, it really isn’t.
The sling straps are made of three strips of soft leather (2cm wide) braided, so it’s a low-density construction with next to no impact if you catch someone with it (not that you should sling at all if someone could be hit on the follow through). The pouch is just a flat piece of leather - it doesn’t need to be formed into any sort of cup to hold the projectile, and the ammo is going to be so light that anything that might get in the way of a sharp release is a bad thing. The knot at the end I’ve covered with discs of light plastazote foam to make a ball about 4cm across, so as not to catch anyone in the eye with the loose end (though see comment above about not slinging at all if you might).
For ammunition, I tried out a lot of different foams - the best performance is a heavy reconstituted foam (with a bit of practice, 30 metre shots are feasible) but I wouldn’t want to hit someone in the face with one of those at close range. Memory foam is quite heavy (as foams go) and exceptionally soft and squishy, and is the best compromise between safety and efficiency I’ve found. The projectiles should be liberally coated in latex, as that greatly improves their flight.
There are a surprising number of different slinging styles to try, but with a light-weight projectile, the simpler ones are best, as if the strands of the sling twist together, the bullet might not have the momentum needed to separate them cleanly.
I started with a straight over-arm cast, which is fairly easy to do, and accurate-ish (at least on the left-right axis - a lot of bullets still went into the ground or over the target’s head), then modified that to a quick over-hand snap from waist height (only really any good at close range). Currently I’m using a side-arm style, with bent knees, standing sideways on, and basically skimming the bullet in a flat arc.
The sling is still slower and less accurate than just throwing the ammunition by hand (and with only a marginal improvement in range), and vastly less accurate than a larp-safe bow, and needs quite a bit of practice to get to the level of being only moderate worse than an unpractised archer or thrower of rocks, so I’m not expecting it to catch on, but I’ve had quite a lot of fun discovering that.