Learn English – When can I “do, do” something

Yes you can, there's no rule against it. Usually it's preceded by a comma:

Why is it that [...], I wonder.

or like this example taken from here, after direct speech:

‘What should I do now?’ she wondered.

As @Mitch comments, whilst it's relatively common in casual speech, I wonder shouldn't really be followed by an actual question (an interrogative).

Taking a typical interrogative, such as who are you, I don't know the specific grammatical term for the reordered version who you are, so I'll just call it a "wh- noun phrase" (which is what you should use after I wonder). Here's the relevant definition for wonder from OED...

(definition #2) Usually with clause: To ask oneself in wonderment; to feel some doubt or curiosity (how, whether, why, etc.); to be desirous to know or learn.

It's particularly worth noting the following supplementary point made by OED...

I wonder is often placed after a question which expresses the object of curiosity or doubt;
e.g. ‘How can that be, I wonder?’ = I wonder how that can be.

...where I've highlighted the word-order switch made in that example.