OK, that's a confusing comment. You can see in th epython code that it is not specifying a path when deleting the file - that means it is deleting the file in the current directory. The word "root" here means the "root of the wiki git repository", I believe.
wget is failing to write to the root of the vagrant checkout. From your searching_Parameters.rst.txt I can see this line:
vagrant@vagrant-ubuntu-trusty-64:/vagrant$ ls -lart
dr-xr-xr-x 1 vagrant vagrant 4096 Feb 10 18:35 .
So vagrant doesn't have write permission into /vagrant. That's... odd.
Can you think of a reason that the "shared directory"
/vagrant is not writable my its owner, the Vagrant user?
You could try
sudo chmod o+w /vagrant, (to give write permission to the Vagrant user), but there must be an underlying reason that it is not writable. Perhaps the directory isn't shared with the host and they don't want you losing changes when the VM is destroyed? If that is the case, why are all the other directories writable? You might check the directory permissions under Windows are sensible, I guess.
FWIW, the directory is writable on the ardupilot_wiki Vagrant VM I just brought up.
Try the chmod anyway and retry running the update.py IMO.