What type of transformer should I use to convert 24VAC to 12VAC (or vice versa)?
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. (a) 12 to 24 transformation without isolation. (b) Fully isolated 12 to 24 V transformation.

Transformers are very flexible and versatile.


  My problem is that I can't find a 2:1 transformer suitable, but it seems impossible that such a thing doesn't exist.


You should be able to find a dual 120V:12V transformer quite easily and use this with the mains side disconnected. If you feed 12 V in on the secondary you will get mains voltage on the primary but you just need to make sure that it is properly isolated. i.e., You don't have to use it.

In Figure 1b I have sketched just one of many arrangements that are possible depending on the transformers available. Here we are paralleling  the 12 V "primaries" and series connecting the "secondaries".

Watch your VA rating overall. Also watch your winding current and make sure you don't exceed the design value.