You cannot run searches like that through the regular NCBI website; that would put too much burden on their servers. Instead you will need to install the BLAST program and download the latest version of a file with all DNA (or protein) sequences in Genbank. Of course that is a huge file, but once downloaded you can run automatic updates.
Here are 2 websites with more information on how to download and run the BLAST program on a server. Be aware though, that these searchers are computationally intensive, and that they can probably not run on a regular computer. You will need to have a powerful server for that.
Alternatively, BLAST/NCBI also offer Amazon Cloud rentals for if you don't have access to a server.
Thirdly, just in case you are looking to assign taxonomy for a bunch of 16S rRNA sequences (I was not sure), instead of using BLAST, I would recommend using the RDP Classifier online tool (super easy to use):
http://rdp.cme.msu.edu/classifier/classifier.jsp
Or the Silva aligner/classifier (also online):http://www.arb-silva.de/aligner/
Just note that Silva and RDP work only for 16S rRNA sequences, not for other genes.
If you're blasting against a specific taxonomy it should be fine to run on a regular computer (assuming you're not blasting against a whole genome/6-frame ORFs or something in which case you need something a bit beefier).
Yes, as Elizabeth mentioned to do such a task would require you to download the standalone version.
How are you planning to get taxonomy specific files? Do you have in-house ones or downloading them from some repository? If the latter, swiss-prot provides some proteome fasta files for many model organisms and will be very up-to-date.