I see nothing dangerous about letting little kids ride in the goat cart, as long as you are walking while leading the goat, the cart has some kind of padded handle in front for the kids to hold onto, and make the kids wear helmets. Also, make the parents sign releases before you even let them on your property, especially if they're customers rather than friends or family members.
Maybe sounds like overkill, but haven't we all met a few parents who require 100% safety at all times for their child, and expect animals to have read the fine print, understand and comply.
ETA: If you're partnering with your friend's pony-party business, make sure you're part of her insurance.
If you don't think it's dangerous, why do you suggest padding and helmets? If nothing bad will happen why would OP need insurance? It's because animals, no matter how small can cause damage and injury. Even you see that while suggesting the opposite to OP. That mindset makes it even more dangerous.
OP, I'm not saying you shouldn't try it, I'm saying it's not as simple as strapping a cart on and walking around. First you have to teach the goat to except the halter. Then it has to accept a harness. Learn to pull weight. Learn to wear shafts, Learn to pull a cart, which means being okay with a "thing" chasing them everywhere they go. Learn to do all that while still listening to the handler. Being in a harness and cart can be very scary to some animals and not all of them are suited to it.
Then it has to learn to do that while not near the barn. While in strange places with strange smells. While people are walking and talking all over the place. While little kids are excitedly laughing or some tired kid is having a screaming meltdown. How about those ponies? Has the goat been around them and isn't afraid? Have those ponies been around goats, or carts for that matter? I've seen horses freak out as we drove past their pasture with my pony and cart. Ever seen a scared horse bolt? Image that in the middle of a kids party.
It's not an easy thing to handle a frightened goat much less one that's strapped to a cart. I have ND so I know that even though they are small, they are powerful. If they get scared it can get real dangerous real fast. Goats are prey animals and, like horses, their first instinct is to run. Image you are running because you got scared but no matter where you go there is a monster right behind you. It wouldn't be very easy to hold onto you at that point, would it? I bet a goat will also try to climb to get away. Think about how much damage/injury could be done by that cart being flung around. Especially if there are kids in it. A goat might be a lot smaller than a pony but kids are small too. The goat could also very easily break a leg or its back if it starts to twist and buck. If a shaft breaks it could puncture the goat.
Like I said, I'm not saying you shouldn't try it. If the goat is suited to it, it would be fun. But if you have in your head that the above could never happen you should stop now. If you deny the risks, you won't take it as seriously and then it's even more likely to happen. You can accept that there are serious risks, and still move forward.