What are the main causes of whether a person becomes introverted or extraverted? Genetics is obviously an important cause of whether someone becomes extraverted or introverted. But what percentage is genetics and what percentage is the environment? What factors in the environment are most important in determining this trait?
Heritability estimates of Extraversion (and other Big 5 factors)

The introduction section of  Loehlin et al (1998) provides a narrative review of heritability estimates of big 5 personality traits (i.e., one of which is extraversion).
A brief extract gives a flavour of some of the research that has been conducted:


  A recent heritability analysis of the Big Five dimensions as measured
  by the Revised NEO Personality Inventory of Costa and McCrae (1992)
  used a total of 660 monozygotic (MZ) pairs and 380 dizygotic (DZ)
  pairs from pooled Canadian and German twin samples (Jang, McCrae,
  Angleitner, Riemann, &amp; Livesley, 1998). For all five traits a simple
  model involving only additive genes and nonshared environment fit the
  data. Estimates for the heritabilities of factor scales for [Extraversion, Agreeableness, Conscientiousness,
  Neuroticism, and Openness] were .50, .48, .49, .49, and .48, respectively. These
  estimates are obviously very similar—in partic- ular, the heritability
  of Agreeableness is quite in line with the rest.


They also mention other twin studies, with heritability estimates for Extraversion, Agreeableness, Conscientiousness, Neuroticism, and Openness respectively as:

                       E    A    C    N    O
Waller (1999):        .49  .33  .46  .42  .58
Riemann et al (1997):  60  .57  .71  .61  .81


Thus, in summary, it seems that such estimates for the heritability of extraversion are around 50%.

Side Point


While inventories like the Myer-Briggs Type Inventory often categorise individuals, into categories, I find a continuous approach to personality traits provides a more accurate description of an individual. Using such an approach, people have a score on an extraversion dimension that is roughly normally distributed (or at least continuously distributed) in the population.


Limitations


I realise this only partially answers your question. It doesn't discuss the dynamic environmental-genetic interactions that cause personality traits , nor does it take a critical perspective on issues related to the measurement of extraversion, and the estimation issues related to heritability. Hopefully, it is a useful starting point.


References


Jang, K. L., McCrae, R. R., Angleitner, A., Riemann, R., &amp; Livesley, W. J. (1998). Heritability of facet-level traits in a cross-cultural twin sample: Support for a hierarchical model of personality. Journal of Personality and Social Psychology, 74, 1556–1565.
Loehlin, J.C. and McCrae, R.R. and Costa, P.T. and John, O.P. (1998). Heritabilities of common and measure-specific components of the Big Five personality factors. Journal of Research in Personality, 32, 4, 431-453. FREE PDF
Riemann, R., Angleitner, A, &amp; Strelau, J. (1997). Genetic and environmental influences on
personality: A study of twins reared together using the self- and peer report NEO-FFI
scales. Journal of Personality, 65, 449–475.
Waller, N. G. (1999). Evaluating the structure of personality. In C. R. Cloninger (Ed.), Personality and psychopathology (pages 155-197). Washington, D.C.: American Psychiatric Press. FREE PDF