There are several ways to set up shared BLAST databases for your organisation depending on what you have available. They are:
1. Geneious "Custom BLAST" on a network drive
As described in the post How can I BLAST to a local copy of preformatted NCBI databases, a local copy of the BLAST executables can easily be set up to search a local copy of any databases you need. This approach can be used to create shared databases by simply putting the BLAST executables and associated data on a network drive.
Each user can then follow the same instructions for setting up Custom BLAST to access the databases. 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.
2. Cloud BLAST
Geneious Prime 2019.1 onwards can connect to cloud BLAST hosted on Amazon Web Services. You must have your own AWS account with cloud BLAST enabled in order to do this. Full instructions are in this post.
3. Raw database files on a network drive
Another approach is to provide users with a set of sequences in FASTA format that they can create a Custom BLAST database from on their own machine. You can update these as required but you'll then need to ensure that users recreate their databases with each update.