To get your image back into a "rectangle" shape, you could use Lens Correction (**Filter → Lens Correction or Shift+Ctrl+R and play with the settings there
Go to Custom and use the Geometric Distortion, Perspective and Scale adjustments.
Then use the Clone Stamp Tool (S) to fill in your edges.
As far as turning your image into vector goes, you can't*.
If you have the image of the dog, you can just recreate the (very simple) design in Photoshop or Illustrator.
*You can try methods such as auto-trace (in Illustrator) but that won't produce the optimal results. The image you are working with is a raster image (the photo) of a raster image (the dog on the shirt) - which makes this very difficult to turn into vector.