By changing "request" in your title to "requested" in your examples, both versions are incorrect. They would have to be more like:
- I requested the picture from John.
(Note no "for").
- I requested John for the picture.
(This is clumsy English and not grammatical).
To use "request", you'd change them to:
- I sent a request for the picture to John.
- I sent a request to John for the picture.
These turn "request" into a noun: the request is the message asking for the picture. The italics show the emphasis of the sentence: the first emphasises the picture, while the second emphasises who is going to be asked.
You can also simply say "I will ask John for the picture."