Will it ever be possible to create a camera that is able to take pictures in the SVG format?

Jason's is a good answer that I'll summarize and add to. Such a camera would capture the image in raster format but then create a vector version in its own computer using an algorithm like to Adobe Illustrator's Live Trace. Similar to how you can get Camera Raw or JPEG from a modern camera, the camera could store a raster and a vector version of the same photo. Instead of tracing the edges of objects and areas in the photo, the camera could create a vector square for every pixel captured. The initial raster capture should be at a really high resolution because of the issue I'm about to describe. In order to resize such a picture, I think it would be necessary to have an app that takes those vector squares and subdivides (for enlarging) or combines (for reducing) them. Reducing is no problem, the app would simply throw away data just like Photoshop does now. It's still not going to be infinitely enlargable, though, because at some level of enlargement, there's simply no real data about the edges and areas beyond what's already in the image, so it will have to take neighboring colors and fill in, which is what Photoshop does already. I'm not sure it would be any crisper if that operation were performed on "vector pixels" rather than raster pixels.

To answer your question let me try to answer some similar questions. For starters, as was stated the camera records in a format known as raster graphics which are of highest quality for recording pictures. Now I will continue with the following disclaimer, I'm not the guy who is going to write a jpeg successor but I do know that converting raster images to vector is a highly computational task which can be done but not with the CPU that we usually find in an off the shelf Cannon or Nikon. from Macromedia Flash 4 and after (perhaps 3 too I don't remember) youve been able to make a vector conversion of various qualities, each forming a distinctive different effect. You might have seen various Twitter and other Digital human avatar portraits that are vector and either hand drawn or converted at a certain low to medium quality. The question shoul perhaps be what is the state of these algorithms and --perhaps, I don't know--is it more efficient to go from analog (lense) to vector, so perhaps your question is a fair one after all. I hope that helps, search for Raster to Vector or vote this up and perhaps I'll do more research into it and post it here.

"...Philip Willis and John Patterson  of the University of Bath in England have devised a video codec that  replaces pixel bitmaps with vectors..." http://www.extremetech.com/extreme/143130-vector-vengeance-british-researchers-claim-they-can-kill-the-pixel-within-five-years

Two parts I can imagine 1. The algorithm can try to join area of similar colours into a vector area. Poor solution though because in real life objects have hard to define texture and gradation instead of the clear numerically describeable colour in vector graphic. If enlarged, you'll get picture like if you run posterize colour in photoshop.   2. The algorithm can try interpreting the dots and make guessing of the image, but as you enlarge the image, the "inflexible" line and curves will show anyway. Moreover like the problem with colours, there's limit of details that can be captured by the lens, and now the image sensor have to guess those too.   Even if possible, it won't work for most photography cases unless limited to artificial stuffs with flat colours and straight lines.

Technically it is possible, but not feasible. All you need to do is to implement a bitmap tracing algorithm in the digital camera. It would be very expensive to develop and would probably need a high performance general purpose CPU instead of the DSPs most cameras use. It would also mean a lot more energy consumption. As I said it’s not feasible because you can simply download your photos to your desktop computer and use a vector tracing application to produce an SVG. Also, automatically traced images are usually not very good, it’s better do it by hand.