I'll be honest with you- if you drive regularly, it isn't a case of IF you have an accident, but WHEN and HOW BAD. There is a saying that any driver who claims they haven't had one is either new or a liar and that's not far off base.
Horses (as you know) are very unpredictable and no matter how well trained, they are eventually going to spook or do something silly and if your wheels happen to not want to go the way the horse does, well-- oops.
Now, that said... there is a lot you can do to prevent accidents.
Take lessons yourself on safe driving practices. Start with an experienced driving horse who will take care of you and learn the right way to handle and harness him. Buy good safe driving equipment, maintain it well, and hitch up correctly. Use a kicking strap. Buy a cart that is stable, comfortable and safe. Build trust between you and your horse so they will listen, train an iron-clad "whoa" and desensitize them to as many things as you can think of. Drive in a sensible manner and don't do anything outside your comfort zone, no matter who ribs you about it. Wear a helmet! Use driving gloves to help grip the reins and protect your fingers. Carry a sharp pocket knife or seat belt cutter at all times and be familiar with what to do in various emergencies. Nothing inspires confidence like being prepared! Listen to your intuition- if you feel unsafe, you probably are. Unhitch immediately and/or get help. I know so many people who got into accidents because they didn't listen to that little voice in their head or heard it but were afraid of looking incompetent. Better "incompetent" than hurt.
If you think something is off on any given day, don't just dismiss your feelings and drive through it. GET OUT OF THE CART AND CHECK EVERY SQUARE INCH OF YOUR HORSE AND EQUIPMENT! They don't act up for nothing- they're trying to tell you something.
I've been in two major driving accidents, a couple of minor ones and one runaway that could have been very bad if we hadn't happened to be in a large field where I could work him out of it. The first major accident was with my Arab and while we took all the precautions I suggest, starting him from scratch to ascertain his training and taking quite a few lessons together, the former owners who said he was broke to drive forgot to tell us about a major accident he'd been in that had totally traumatized him. When he took that cart off the trainer's property for the first time he flashed back without warning and panicked. He never drove again after that.
The second bad accident was with a nice little mare I was borrowing and I'm not sure I'll ever know what caused that one. Did she get stung by a bee? Did something spook her? I don't know. She took off without warning and we probably would have been fine except she kicked my foot off the roadster stirrup which pitched me out of the cart. If I'd used a kicking strap as I usually do that wouldn't have happened and I'd probably have gotten her under control without incident so I put that one down to failure to use proper equipment.
The minor incidents stayed minor because of all that training I'm so firm about.
In one case I ran the cart too close to a tree and my EE tipped over on a root but I held onto the reins and said "Kody, whoa!" and even though he was upset he stopped. Shaky, nervous, twisted up in the shafts, he still stood there and waited for me to come get him out of it. Yay for trust! Neither of us had so much as a scratch; I just righted the cart, straightened his harness, gave him a scratch and praise and we drove on. I can't tell you the number of times that sort of trust and training have prevented problems for us.
What you need at your stage is a nice calm older horse, maybe even a bit of a dead-head, who will take you down the country road at a nice steady trot and build your confidence. It's true that you can get hurt just as badly driving as you can riding, but the risk is greatly reduced with miniature horses simple because their equipment is so much lighter and the distance to the ground so much less. If you wear a helmet and use common sense you'll probably enjoy many years of safe and happy driving. It's a wonderful past-time and well worth the risk!
Leia