How can I create a rearrangeable list of items like OptionTree offers with the Theme Customization API?

I suggest you take a look at the Kirki toolkit: https://github.com/aristath/kirki
It has a “repeater” field that should do what you need.

It’s still under heavy development and for the time being I’ve only added the ability to use a few field-types with it, so I guess it depends on what exactly the data you want to add & re-arrange is, but it’s worth a look.

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