There no shame in using raster images. You should be aware of that. A high quality, high ppi raster image is exceptionally useful in many instances. That particular image is problematic due to the low contrast background and vignetting. But a clean drawing scanned to a clean image would be more than sufficient for many methods of output.
If you really want vector, then Joojaa is correct in his answer.. redrawing is the best method.
With experience you learn to draw with vector conversion in mind - often using less detail in the drawing knowing you'll add detail in the vector artwork. It sort of gets like tattooing... you draw basic shapes, scan and manually trace those, then add all your detail in the vector application rather than by hand. This saves doing the work twice.
Barring that, tracing is pretty much the only other option, short of hiring someone to do it for you.
Different software will trace differently. You may need to experiment with different auto-trace features in various application to find one which you feel works best for you. In general, you scan as large as possible, use a raster editor to make certain the contract of the scan is high, eliminating any subtle grays and making certain blacks are black.
Essentially you have to learn to scan and adjust for the trace... most tracing features will work best with art as large as possible allowing them to see more detail.
There's no simple solution overall. I, personally, would not be able to do anything with your sample image other than redraw it due to the background and low contrast.