Most minis just use easy entry. Some (mainly the beginners) people with ponies and horses do as well. Most ponies/horses who've been in it for awhile use marathon carts as they're safer to use for the marathon/hazards portion. VERY difficult to find a well made mini marathon vehicle, although there are some. They're also VERY pricey! I got a SUPER AWESOME deal on mine as I bought it with my pony from his previous owner ;)
As far as equipment, you need a good safe cart (Frontier type easy entries work fine for minis) and a well made harness. I prefer betathane to leather, hate biothane. Beta is nice because it's easy to care for and all the water and mud you can get on it during the marathon won't bother it. There are some really nice sport harnesses out there, such as the one Chili has (complete with the spiffy Freedom collar). I'm drawing a blank on the name of it...
Oh yeah, there are rules regarding wheels. The "bicycle" type aren't always allowed. Our club allows them for all levels in ADTs (see below) and for training level minis in the CDE. But above that you have to have the non-wire spoked tires for safety. I borrowed a pair to compete my mini in prelim at our CDE. Our club only has one CDE a year, but 6 ADTs (and some darbies) so for the CDEs I usually just do my pony. One of these days I'll get some of the accepted wheels, but they're expensive and I just haven't gotten around to it.
You need to dress appropriately for dressage and cones, as presentation is part of it. You need reflectors on your cart for marathon, as well as a safety triangle (those might have another name?) which are all inexpensive. Really, the best thing is to check out the two websites I posted, they'll tell you what you need to know :)
My driving club puts on ADTs, but they're not the typical ADT. Normally it stands for arena driving trial, as they're held in arenas. But ours we call Arizona Driving Trial, because ours are more like CDEs, but they're less formal (so no special dress, no triangles needed, ect) and instead of a marathon you just have some hazards, or maybe a SHORT marathon that isn't timed like a CDE would be. For a CDE, yes you do need them in shape (goes for ANY size horse) as, well, it's a marathon ;)
Indy: Yup, that's my Fell pony, Harley :) His prev owner did CDE with him, as well :) My Fjord will be doing them in the near future, but right now he's very green so I sent him to my trainer (she gets them trained better and quicker than I do lol!) My goal is to have him in our CDE in March :) Anyhow, ALL sizes of horses compete in CDE. There are the VSE (very small equine) which is the minis, then there's ponies and horses. Our club has had sizes from 29" mini up to Belgians (maybe even larger? Not much larger than Belgian workhorse though haha). Also donkeys and mules!
When I have some time at home I will post some pics of other competitors. Hubby and I love to take pics in between our times if we can of others, so I have a good collection. I'll post it in the photo section, but link it here when I do.
I HONESTLY say it's the MOST fun I've EVER had with a horse!