Pokemon-go – Why can’t I add Pokémon to the team’s gym

The most likely cause of this is that all of the gym's slots are full, which means you will not be able to add another Pokémon until the gym is upgraded. Gyms can hold a number of Pokémon equal to the gym's level, so the gym in question is likely only level one, and its slot is already full. To increase a gym's level, you must win friendly "battles" (training) against the gym using only one Pokémon (as opposed to the usual six). Every time you beat a Pokémon at that gym, its prestige will raise, eventually leveling it up and adding another slot. At this point, you should be able to add a new Pokémon to the gym.

As Riley mentioned, the button will appear in the bottom left corner of the gym when there is an empty slot, and any Pokémon that you wish to place in that gym must be full health.

This happened to me a few days ago. I was with 4 friends and only one space in the gym (we're all in the same team) and only one of us could put a pokemon in the gym. It turned out to be, that the game I think has a bug that only enables the first person who entered the gym to add pokemon and so on.