There are two different types to consider here. One is the "miniature" and the other is the full size "pull type" wagon. Of these two the least common and less seldom thought of is the "miniature".  Thousands of miniature wagons were produced as cast iron playthings that may or may not have accompanied other toys such as dolls and tractors. Then there were the many miniature wooden wagons produced the very early America. These items are not nearly as common as the full size pull type wagon that most think of, yet many of these miniatures can be quite valuable as collectors items.

As far as antique pull type wagons, the best known by far is probably the Radio Flyer (the original "little red wagon") and yes the company is still in business and selling wagons today -- see RadioFlyer.com for the company's current website.

Short History of Radio Flyer - and the toy wagon of course
Antonio Pasin, a skilled craftsman from Venice, made his way to Chicago where after several dead end jobs he managed to save enough money to start his own business. He worked at night building wagons and sold them during the day. His company was originally know as the Liberty Coaster Company. His first and most famous wagon was called The No. 4 Liberty Coaster and it was handcrafted entirely of wood.

The company was later named the Radio Steel Company. It was at this point that the Radio Flyer (made of steel) was born. Through the years Pasin tinkered with and produced a few other products (gas cans for the military, scooters, row carts and even outdoor furniture) but his focus and detail was always on the wagon which became basically synonymous with Radio Flyer. 

Radio Flyer Was NOT the First Toy Wagon Maker!
The antique toy wagon is ancient -- literally! Miniature toy carts were found in the tombs of Egyptian pharaohs. Some wheeled toys have even been found in the burial grounds of native Americans who never built or used wagons.