Primer-BLAST is a tool for designing target-specific primers, available through the Primer-BLAST web interface.
Is it possible to run the tool locally instead of accessing it via the internet?
Primer-BLAST obviously uses NCBI Blast which we all know about, but it is a pipeline running through Primer3. This is an extremely popular and reliable primer design software, but only does single stranded DNA/RNA. There is a prettier version, Primer3plus here.
To strictly answer the question its not possible because it is a pipeline based on Primer3 which is not written by NCBI, therefore they would not be in a position to distribute it. Nor would Primer3 be able to distribute Blast because that is NCBI. It is simply a web-server.
NCBI and Primer3 would need to come to a joint agreement, which they must have done to present the webserver (at least it would be polite). You can of course replicate the pipeline yourself.
Brief details are here, https://www.ncbi.nlm.nih.gov/tools/primer-blast/primerinfo.html
It can be done via Docker. This exact pipeline hasn't been distributed via Docker but Blast is available as a Docker image.
New Answer Just to follow this up, primers4clades is a Docker image aimed at very similar objectives to Primer-Blast. I have not used it, but its worth considering if standalone is essential, e.g. for ease of building a pipeline.
Here's the blurb
Primers4clades provides a fully automated pipeline for the design of PCR primers for cross-species amplification of novel sequences from metagenomic DNA or from uncharacterized organisms belonging to user-specified phylogenetic lineages. It implements an extended CODEHOP strategy based on both DNA and protein multiple alignments of coding genes and evaluates thermodynamic properties of the oligonucleotide pairs, as well as the phylogenetic information content of predicted amplicons, computed from the branch support values of maximum likelihood phylogenies.
I don't know the user, but Docker is a safe running environment that is isolated from the system. There's no harm in checking.
Finally both Blast and Primer3 are available via conda, but not Primer-Blast. In theory, I believe Primer-Blast could be distributed via conda under the rules of conda. In fact, I think, but could be wrong, that anyone could rebuild this pipeline and post it back to conda. @merv will know exactly whether this is 'permitable' and is very likely to correct me if I'm wrong.
