The correct solution is to set up a WWWBLAST NCBI mirror locally and mirror all the BLAST databases as well as add some of your own. This will replace access to the NCBI service itself though. If you are unable to do this you may be able to use Custom BLAST to achieve something similar.
One approach is to provide users with a set of sequences in FASTA format that they can create a CustomBLAST database from and keep that up to date and have them replace their local copies. This has the advantage that it is essentially purely parallel so it will scale indefinitely but it has the disadvantage that you can't be sure they're all searching the same database.
Since the Custom BLAST service access a folder on the user's hard drive, it is possible to put this folder on a share and have each user point at it. Their CPU will do the work but that data will be centralised. It is possible that this could cause performance issues over the network though and you'll need to deal with ownership and ensure that your users don't try adding databases themselves.
It is also possible to create BLAST databases outside of Geneious, by running the BLAST+ executable from the command line and using the makeblastdb command. Then put the formatted BLAST database files into the BLAST/data folder where your other custom BLAST databases are located and Geneious users will be able to see them when they run custom BLAST. Note that if the database is formatted manually using makeblastdb, there will be no annotations on the resulting alignments. If it is formatted from within Geneious, then an extra file is created with the annotations so Geneious can put them back onto the alignments after a search.
See also this post for more information on BLASTing to local databases.