At that temperature, decals will suffer, especially if humidity is also high. While 120°F will not melt, or even significantly soften styrene, prolonged exposure to such temperatures will cause it to become progressively more brittle, especially clear parts.
There are ways to help keep them cooler: Make sure that the shed has adequate bottom to top venting. Cooler air comes in at the bottom and hot air exits at the top. Don't store the kits near (3 feet) the roof. Stack the kits on a pallet or something like it to promote air circulation around them. If possible, put them in an insulating container, even just surrounding them with sheet styrofoam will help considerably.
The biggest problem may be humidity and it's effect on the cardboard boxes. Cardboard is great at absorbing and holding atmospheric moisture—and growing mildew and mold. Some molds will grow on the plasticizers slowly migrating from the styrene, too. This can occur even in relatively arid environments, since enclosed structures, especially with slabs-on-grade, gravel, or tamped earth floors, tend to draw and concentrate humidity. (See comment on venting, above.)
Lastly, don't store any petroleum product, or machinery that uses it, in the shed. Gasoline, even in vapor form, is hard on styrene, especially in a closed, hot environment.