Look into a mini Meadowbrook - all wood construction, wooden wheels. Won't get hot in the AZ sunshine, won't chip paint, very solid and sturdy....and looks cute...and authentic. And won't break the bank to buy it. Lots of sites for them - just google.
There are a lot of posts you can search about this topic.
Lots of opinions...there's the metal cart/Jerrold crowd (bike wheels, pneumatic tires etc.) They are light, don't balance well and spoke wheels are dangerous when a side load is applied.
Anyway, I like meadowbrooks, but they're too heavy for a small pony. They're also tough to get in and out of. You're trapped in a tub.
A nice easy entry with wood solid tired wheels is great and light. Buy used if you can. Look on Ebay, lots of carts. Also, go onto the CDL (carriage driving list, not truck driver) forum/list...great info and people always have somethign to sell.
If the Meadowbrook strikes your fancy, but you hate the egress problem - we ALL do, btw - then get a wooden road cart with front entry. Just make sure it has the seat backs!! A 'no seat back' cart can leave you sitting on the road, watching your pony head into the sunset, if you get flipped out from a spook or bolt or a wheel going over a big rock or any other silly thing that can happen to your little cart while driving.
A simple seat back will prevent that embarrassing type of incident. It will keep you in the seat, and in control.