I don't have the same model VAIO that you do, but it does sound like you have a broken fan. The system probably has an override that shuts down when it gets too hot to prevent damage.
You can disassemble the box and look at the fan. This will be a bit of a pain because there will be many screws to remove. There are usually arrows pointing to the screws, which helps (remember to check under the feet!).
Alternatively, there are laptop bases that you can buy which have built in fans and some type of thermal material to keep the bottom cool. These are generally for laptops that get too warm to put on your lap, but it may help if you can't find replacement parts.
Good luck!