Is it possible to recreate this sort of blurred gradient in an SVG?
Yes and no.
It's possible to create a gradient mesh in Inkscape, and save it out as an SVG. However the SVG standards used in browsers don't support gradient meshes yet. This might change in the future, but for the present no.
Another method which does work in browsers, is to add several solid objects, blur them, and put them in a clipping mask. This is still a pure vector solution, it's not rasterized.
Here's the SVG for the version that works in browsers, if you want it.
Edit: the above link to the SVG is no longer available, so I've removed it.
It can be made in Illustrator and Inkscape as Gradient mesh. When saved as SVG in legacy Illustrator, it transformed to bitmap image. Obviously Illustator knew it about 6 years ago, but SVG language didn't.
When saved in Inkscape as plain SVG, not as Inkscape SVG, it stayed as gradient mesh and opening the SVG file with a text editor revealed there's command meshgradient. See the code snippet
So, without knowing the standards, I can say "It's at least implemented in Inkscape's plain SVG."
Hopefully someone SVG development status aware programmer tells something exact.
If your goal is to have a lightweight, scalable mesh gradient then you could try this method:
It’s not vector but it might as well be since the file size is super small and the image scales infinitely.