Should button values be translated?

Upon rereading your post, I think I'm going to answer "I don't know." Here's why:

My question is: Can I translate the button value in InDesing to "Sí", so that it would display in Spanish in the interactive PDF? Or would that break the radio button?

You can set those values to whatever you want, in either Acrobat or Indesign. However, when you're working in a form with some interactivity, I don't know if we can know what consequences there might be for changing that label. I mean, if I were designing an interactive form for my state org translation client, I wouldn't design the form that way, but I would be entirely unsurprised to encounter a form used the button value in some other scripted action. So changing "Yes" to "Si" might break something else, besides the radio button, but the only way to know for sure would be to Try It And See.

(Or to read all the JS embedded in the buttons, looking for something with

this.getField("TypicalFieldName").exportValues

... which is how I'd personally grab the Yes/No values off of a radio button with JS in Acrobat.) 

If you set up a radio button in the Buttons and Forms panel, there's a "Button Value" field at the bottom of the PDF Options section. That's where you can put your translations of the Yes and No values.

In Acrobat, the "Yes" and "No" are found in the Options tab of the Radio Button Properties dialog. So if you were doing 'em manually, it'd look like this:

Correction: In the translated PDF, it appears as "Quiero recibir avisos de audiencia por parte de la División de Audiencias Estatales por correo electrónico: Yes"