How to decide what joints to use for a frame of a play wagon

Let's capture some of the comments in an Answer.

For this application almost any joint will do. Though, riding toys are subject to forces that might warrant putting on our engineering caps.

The idea here is that the rails of the wagon are like rails on a deck. When force is applied at the top of the railing this translates to a lot of force at the joint, where the rail joins the decking. There are a few ways to guard against this, and the joinery used is actually not what is going to make the largest difference.

Unless you are going to integrate the railings directly to the deck with some fancy cabinetry joints, you are relying on mechanical fasteners anchored in various types of wood grain. This sort of connection will eventually fold up, probably quite suddenly. This might surprise a 9-year old rider inside the wagon.

So, what we want to do is find a way to have these railings support each other, so they are better able to handle forces where they are weakest. This is why I think the design tries to tie the railings together.

Think about how your fasteners are weakest and strongest, and work within these parameters.

I will point out that the traditional wooden little red wagon is built a lot different than we see here. Typically, there is a butt-joined or mitred box with a rail down the middle that supports the bottom and carcase pieces that form the basis for attached the axles. Often there are metal or wood corner cleats. This stiffens the box to reduce racking and separating. Then the rails are often removable sections that drop into metal or wooden slots on the outside. The strength of the wagon relies on a stout box that the rails attach to, but they only have to resist the forces against them, not be the entire side support.

Lee Valley has an example of what I'm talking about, but this is a very common and time-tested design. It can be found any number of places, and is based on a very typical design I'm sure you can find references for online.