A box full of scorpions
The first step is to wear apropriated gloves. Grabing a scorpion with your bare hand is dangerous, so you should use gloves to avoid being stinged and poisoned when handling them.
If it is a large bag or box with thousands of scorpions inside, most of them would poison their peers and other would be suffocated. Further, they would be hurted by the movements induced in the bag or box when the skirmisher is walking, running or fighting and be shaken and crushed around inside the bag. The result is that when the battle begins, most of the scorpions would already be dead or very injured and the progress of the battle just makes things worse. Further, even if they happen to survive and be in good health, the surviving ones would probably be moving frenetically in panic trying to leave the bag or box. Putting your hand inside a bag or a box full of living scorpions trying to escape in panic don't seems to be a good idea, even when wearing gloves. So clearly, a simple bag or box for carrying the scorpions won't do. The skirmishers would need to bring specially constructed bags or cages for the scorpions.
So, the scorpion tamers would need to have an special bag to carry the scorpions in the battle. Ideally, this bag would be a box designed to protect the scorpions from impacts, shakes and crushings induced by its wearer walking, running or fighting and also to protect each scoripon from each other scorpion. Also, the scorpions should not suffocate. This could be achieved by making it a rigid cage (so it would not deform significantly) with a cell-like, grid-like or beehive-like structure, where each scorpion is completely immobilized in its own cell. The cell is internally coated by some soft material. Further, each cell has a small hole to the outside to ensure air circulation and should be designed in a way that the skirmisher should be able to safely, easily and quickly release and grab the scorpion.
It would be something like the following picture:
The inner walls (pink) and the inner floor (yellow) would be made of some soft material to absorb impacts that could injure the scorpion (red). The front wall has a hole to ensure air circulation. You can see that the scorpion is immobilized by the gray wires that act like a seatbelt on them and restrain their movements. Also, the gray wires have some supports against the cell's floor to avoid crushing the scorpion. The wire are held firmly by the two blue scrows. This box is open in the top and the reason is because it would work like a matchbox. Each cell would really be two boxes, where the outer one has the bottom, top and the two lateral sides but is open in the front and back, while the inner box is open just at the top. With this design, to grab a scorpion, you just get the box, open it just like you would do with a matchbox, and grab the scorpion by getting its tail with your thumb and pointer fingers while bending and breaking the gray wires.
The size of the great box would depend on the number of cells present, and hence, the number of scorpions. Sometimes, larger boxes would be desired over small ones if the skirmishers would need to entrench or go into missions were they need to take a large number of scorpions. Sometimes, smaller boxes would be desired over larger ones because the skirmishers would need to go to battle in a way where saving weight is important and there is a cart full of scorpion boxes nearby. Also, some soldiers might personally prefer to work with smaller boxes while some others would prefer larger ones.
Opening a matchbox is very easy to do. It would need some practice to appropriately grab the scorpion from the matchbox in just a few seconds without hurting it or accidentally dropping it on the ground or anything like that (and please, do not forget the gloves!) Anyway, this is something that anyone can possibly master even with only a couple hours of training.
Throwing scorpions with your bare hands is realistic only in a very few handful cases. Might be useful when there is a wall of shields marching towards your army. When they are close enough, your soldiers throws the scorpions over the shields, and immediatelly attack using whichever weapons they have in their hands, possibly running backwards to get a few more seconds in order to allow the scorpions sting the enemies.
Scorpions aren't very aerodynamic and will present a lot of air resistance when thrown out. With this, no one would likely be able to throw a scorpion farther than something like 5 to 8 meters. If you use a sling, you might be able to reach something like 10 meters. If you attach a small stone to the scorpion and throw the stone with the sling, you might reach 20 or maybe 30 meters, but this will likely also kill the scorpion.
Attaching it to a javelin that somehow release the attached scorpion or scorpions only when it hit the ground is probably more effective, since they might land softly enough to allow at least part of the scorpions survive the trip and then detach from the javelin to walk and sting around wherever they landed. This has the advantage that even if the enemies protect themselves with shields from the javelins, the released scorpions might end walking over the shields into the soldiers hands or fall into their heads, destroying their morale and making them shake and scream in panic instead.
For a hand grenade, it should be done of a material that absorves most of the impact while releasing the scorpions alive. I can't think of a material that have the right properties and not be something high-tech. The best that I can think is a ball of mud full of scorpions that would splat when hit the target with at least something like a third of the scorpions surviving. However it can't be prepared prior to the battle and stored because the scorpions would either die from suffocation inside the mud or escape, and this makes it completely unsuitable for any serious use as hand grenades.
Catapulting individual scorpions against an army is also likely to be useless. If you already have a catapult, then you would be better to simply throw stones instead. However catapulting a mud-grenade or a cow carcass full of scorpions over a wall in a siege might work as a way to frighten the enemy and luckly make some of them stinged.
It might be interesting in arrows. The scorpions could be already glued to arrow tips in their box by theirs stings, and then the archer would remove the arrow tip from its matchbox, break the scorpion tail in order to keep the sting in the arrow while removing the rest of the unaerodynamic scorpion from it, then attach the arrow tip to the rest of the arrow and throw it with the bow. The result would be a poisoned arrow.
Also, scorpions might be useful to be thrown at the enemy when defending a fortification or stronghold against battering rams or enemies trying to dig under your wall. Normally, rocks, or boiling water or oils would be better, but scorpions will be more effective against their morale.
Accordingly to wikipedia:
Sting and venom.
All known scorpion species possess venom and use it primarily to kill or paralyze their prey so that it can be eaten. In general, it is fast-acting, allowing for effective prey capture. However, as a general rule they will kill their prey with brute force if they can, as opposed to using venom. It is also used as a defense against predators. The venom is a mixture of compounds (neurotoxins, enzyme inhibitors, etc.) each not only causing a different effect but possibly also targeting a specific animal. Each compound is made and stored in a pair of glandular sacs and is released in a quantity regulated by the scorpion itself. Of the 1,000+ known species of scorpion, only 25 have venom that is deadly to humans; most of those belong to the family Buthidae (including Leiurus quinquestriatus, Hottentotta, Centruroides and Androctonus).
If they are meant to be throw by javelins or grenades to survive the travel and sting someone, smaller species are preferable. If they should be very visible to the enemy to lower their morale, then larger, easily visible scorpions are better. Also, it is easier to handle larger scorpions than smaller ones.
For the larger scorpions, a Deathstalker (Leiurus quinquestriatus) might be what you want, it is one of the most venomous scorpions and is large enough to be handled by your soldiers. Centruroides bicolor might also be suitable since it is large (but the power of his venom is still poorly studied). Hottentotta tamulus is also deadly venomous. Also, they color are quite distinctive to fright the morale of the enemy. If you want something black for nightly attacks, try Androctonus bicolor.
Also, see SeanR's comment:
If I absolutely had to weaponise a scorpion, it would be Parabuthus transvaalicus. Not the most potent but:
"LD 50 value for this species is reported to be 4.25 mg/kg. Due to it size, this scorpion can inject very large amounts of venom. NB! This species is able to squirt venom up to one meter away, and venom in the eyes can be very dangerous."
Also, accordingly to wikipedia, Parabuthus transvaalicus has 90 to 110 millimeters, which makes it a big and black or dark brown scorpion. Very good for nightly actions!
For small scorpions to put on javelins and mudballs, sorry. I don't know which species would be suitable. Centruroides suffusus might be suitable. Many species are not well studied and small scorpions with 4 cm or less are easily overlooked even by biologists.
Scorpions vs men and elephants
Scorpions work best against unarmored or lightly armored enemies. Against strongly armored enemies they are ineffective, except perhaps against their morale. They might work well with elephants, since their skin, although thick, is sensible to insect bites. However, a single scorpion will not kill an elephant because they are too big and massive for that, but they surely can be disturbed when stung.
This also means that your army would need to be enoughly armored to avoid that any accidental release means deaths and morale problems between your own men. Also, they should be trained about how to quickly kill escaping scorpions if needed and how to properly react if they let some scorpion accidentally escape.
Overall, I don't think that using scorpions would be an effective tactic. The scorpion does not have enough brain to be trained to don't attack the tamer but attack anyone else. Also, as noted in Steve Bird's answer, scorpions that are less likely to attack the tamer would also be less likely to attack the enemies and even the most agressive species might chose to flee instead of attack. Also, since you would need thousands and thousands of scorpions, taming then one by one would be expensive. So, the best is to farm an agressive and venomous specie and delivery them immobilized and untammed in the matchboxes.