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To get rectangles of data, use SPARQL queries.  SPARQL is a simple query language designed for use with triple stores.  A collection Use of SPARQL queries used to answer real-world questions is available is described here:  SPARQL Queries

Getting Graphs of Data

The entire triple store can be unloaded for use in a local triple store, and for local query.  This is recommended for sites wishing to make repeated analyst queries of the data.  Community-editions of a triple stores are available with no cost.  Stardog is a popular, stable, and free triple store that can be used for this purpose.  See http://stardog.com

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To get data from VIVO on a regular basis, you may wish to work with your VIVO providers to create an API that you can use that will provide required data.  The Data Distribution API is designed for this purpose and can be configured to return specified data, including parameterized data via configurable addresses in configurable data formats.

Distributed Query

Some applications involve getting data from multiple VIVOs.  VIVOs running version 1.10 and above provide a Triple Pattern Fragments endpoint which can be used to rapidly get all triples from a VIVO matching a triple pattern.

References

1. Börner, Conlon, Corson-Rikert, and Ding (eds) VIVO: A Semantic Approach to Scholarly Networking and Discovery, Morgan-Claypool Publishers, 2012.  160 pages.
2. Allemang, and Hendler.  Semantic Web for the Working Ontologist, second edition.  Morgan-Kaufmann Publishers, 2011.  354 pages.
3. DuCharme. Learning SPARQL: Querying and Updating with SPARQL 1.1. O'Reilly, 2011. 235 pages.