Ball manipulators are one of the hardest mechanisms to build in VEX robotics. This is due to the many factors that affects how balls fly across fields. The VEX vision sensor could be used to improve accuracy, if the maths and physics is too much for you.
"Flywheels throw game pieces by having the game piece come in contact with a spinning wheel. These wheels can be aligned in pairs in which a game piece is spun out from between the two wheels or a flywheel can also be assembled with a single wheel combined with a friction plate which the game piece rolls along. Flywheels spin at a very high speed in order to throw game pieces." - VEX
In order to build a flywheel, two factors must be considered. The angle in which the ball shoots and the speed. The physics / maths of all this is written here. A flywheel shooting a Nothing But Net game ball from one side of the field to another could reach at around 45 degrees angle and 3000 rpm speed. The rpm (rotations per minute) of the contacting wheel can be multiplied by using gear ratios. To learn more about how gear ratio affects speed and torque, watch this video.
There are also two types of flywheel, one which contacts the ball from 2 sides with 2 wheels, as shown in the first photo, and the second which only uses one contact point as seen in the second photo. The choice should be made depending on the space available for the flywheel and considering the ball intake and lead up to the flywheel.
Here is a flywheel explanation that uses ratchets.
"The game piece is placed on a game piece holder and is mounted to a Linear Slide Track which has a series of Rack Gears mounted to the slide. The motor rotates the spur gear and it moves the Linear Track assembly back stretching the rubber bands/latex tubing of the slingshot." - VEX
The slingshot is the easiest and most beginner friendly ball manipulator to make. There is little need for speed tuning like the flywheel and the angle can easily be adjusted by changing where the puncher is set on. As the motion of the puncher mostly relies on the strength of the rubber band stretch, it is important to keep in mind to change the rubber bands to keep the tension. Making the puncher usually requires a slip gear, which is a gear that has some teeth shaven. This will allow for the puncher to work with the motor only having to turn one way and allows for continuous shots.
"Catapults throw game pieces with a lever arm. The game piece is placed on one side of the lever arm in a game piece holder and elastic bands or latex tubing are attached to the other side of the lever. One way to assemble a catapult is to use a Drop Off Cam from the Advanced Mechanics and Motion Kit. A motor will rotate the cam which pushes the lever arm down and stretches the rubber bands. When the cam reaches the drop off point, it releases the catapult flinging the game piece. The cam can continue to rotate and repeat the cycle. A pneumatic cylinder system can also be used to fire a catapult." - VEX
The method that VEX suggested of building the catapult using a Drop Off Cam is quite hard to achieve seeing as they are unavailable in the VEX store and is hard to get. So in order to make a catapult, the gear teeth has to be cut off the gear just like the puncher. Catapults takes quite a lot of space although not as much as the flywheel as it has to have the space leeway to go backwards and forwards.