There's numerous solutions to lifting game pieces for pick and place games, the following is a presentation on the most common methods
There are quite a few consumer off the shelf options for lifting mechanisms, here are some of the most common.
This is a telescoping mechanism that is a kit for assembly, the team used this in 2023 for their telescoping arm. We did have to do a lot of problem solving and modifications to make this arm work for horizontal loads because it's designed for vertical movement. Climber in a Box
This is a kit similar to the andymark kit but it consist of a lot of 3D printed parts. Thrifty Telescoping Tube