They would be very helpful.

Not my pieces, but this is the same vintage wagon I have:



You can see the flat brass end just in front of the screw that holds each horse together.

This next picture shows the silver shaft of whatever the connector is, going through one horse, the double tree, then through the second horse.



The auction I stole these images from is at: https://www.ebay.com/itm/Vintage-Cast-Iron-Beer-wagon-and-8-horses-Nice-piece/164610553314?_trkparms=aid%3D1110006%26algo%3DHOMESPLICE.SIM%26ao%3D1%26asc%3D20201210111314%26meid%3D5d963be5c7164eaeb524bb7006aa8b6b%26pid%3D101195%26rk%3D4%26rkt%3D12%26mehot%3Dpf%26sd%3D233893151926%26itm%3D164610553314%26pmt%3D1%26noa%3D0%26pg%3D2047675%26algv%3DSimplAMLv5PairwiseWebWithDarwoV3BBEV2b%26brand%3DUnbranded&_trksid=p2047675.c101195.m1851

thanks.
Rivet seems likely, but test if one of the others might unscrew. I've seen custom bolts that have screw on heads or thread in the middle from this era.

A quick solution might be to find a couple of nails with similar head shape and enough length (or can be cut to) to meet in the middle. A bit of epoxy to hold it and a dab of brass paint to match the others.

IF that hole is 3/16" or larger that might be the first thing i'd try. they come in varying lengths and the cheapest/easiest to procure are at the office depot. different finishes and materials are available but that 3/16" diameter is constant.

https://www.binding101.com/binding/binding-supplies/screw-posts?msclkid=a2c2c71cba5716ab08daa737a319ab21&utm_source=bing&utm_medium=cpc&utm_campaign=NX_NTM_Search_Binding%20Supplies%20(L)&utm_term=chicago%20binding%20screws&utm_content=Screw%20Posts%20-%20Aluminum

carriage bolts come in every diameter and length at your local hardware store. some thread might show between horses but that could be covered by running through tubing stock or even heat shrink tube.

sort of puts me in mind of a "20 mule team" model i had as a kid.

Some of the eBay auctions mention a manufacturer, but it was not important to me, so I didn't make note of it.

Maybe look up some of the names suggested here:

https://en.m.wikipedia.org/wiki/Sex_bolt

They would do the job but I can't find any with flat heads and without even a slot for a screwdriver. I'll hunt some more with that as my search parameter.

I would bet you could make something with some sort of hollow tubing or long ferules that you could use a screw in, or even some kind of epoxy type of a situation.

Of course I would use bailing wire.

But I'd really like to emulate the originals even if I'm not doing a full restoration.

I'm thinking the easiest thing to settle on would be a sex blot aka Chicago screw. I had a whole stash of short ones that were melted down when the barn burned. If I use the male ends where visible and used some sort of tube the right size to screw (or glue) them into, that might work.

Another option would be binding posts like are used to hold big scrapbooks together. You can get extensions to lengthen them and one end is flat, while the other end has a very shallow slot to screw them together. Problem is, they don't often come in brass - but I could paint them brass color. I might even have some of those, if I can remember where they are.

I've used binding posts a lot with scrapbooks. They come in different lengths and you can get extensions to make them longer. I know I don't have any brass ones, but if I can find some brass ends, I can use the extensions I have in the middle.

OOOH! McMaster-Carr has brass binding posts in the lengths I need! They're a bit expensive, but they would be as close as possible.
https://www.mcmaster.com/brass-rivets/brass-low-profile-binding-barrels-and-screws/

I measured the shafts of the fasteners on the other horses and the shaft is 1/8" in diameter with heads 1/4" in diameter.

This makes it much more difficult. McMaster-Carr's smallest binding posts diameters are 3/16" which might possibly fit, but I am not willing to pay for them on the off chance. Plus they don't have plain flat heads, they have Phillips/flat slots.

So my next idea is to find brass nail head double cap rivets and a shaft (piece of heavy wire or thin rod) that will fit between and drive the rivets onto the wire or rod. Although in the original the heads on the sides are brass, the shaft between is gray color, some generic metal.

Since the double cap rivets are used in leather work, I may call a local shoe shop and see if they have them. Or, my husband knows a guy who makes belts - he probably has them by the hundreds and I only need four - unless I replace all the same parts on the rest of the team for the beer wagon, then I'd need eighteen. Sixteen for the eight horses on the beer wagon and a pair for the horses on the ladder wagon.

OK, I found some "rapid rivets" at Tandy Leather with the right size heads, 100 for $3.99 that might work. That's little enough money to try them out!