It sounds like she needs more practice standing patiently in different environments. Practice means you stand next to the cart, holding the reins, and say "stand." If she starts to walk forward, you pull back on the reins and say "stand."
We like to use "whoa" for stopping after you're already moving, and "stand" as the command to stand still and wait. For such a young horse, I'd practice walking the horse a few feet and then stopping and saying "stand" again. Make her stand patiently for several minutes at a time. Doing this consistently will reinforce that the cue is solid. A horse needs to be doing this well not only for getting into the cart, but also in order for the driver to safely get out.
Unless you have someone to hold the horse's bridle when getting in and out of the cart, I wouldn't drive the horse until you have a more solid response to standing and waiting.
Be very careful and I'd suggest returning to the trainer for any small issues that come up. Horses with carts attached are even more dangerous than ridden horses, so if you're new and the pony is also young and green, you can quickly and easily run into trouble. Even just having the wrong response to the horse spooking at something on the road can prove very dangerous.