1. Are extraverts more talkative than introverts in oral communication?
Yes they are. The (probably) best evidence can be found in research by Mehl et al. (2006) who "tracked 96 participants over 2 days using the Electronically Activated Recorder (EAR), which samples snippets of ambient sounds in
participants’ immediate environments". The correlation of extraversion with word count was .29, indicating that extraversion was related to more talkativeness.
2. Are introverts more talkative in written communication?
It does not seem so. A study that addresses your question has been done by Gill and Oberlander (2002). They asked participants to compose two emails to a good friend. The more extraverted they were, the longer were the emails (beta = .20). Thus, this study was not in line with your intuition that introverts are more talkative in written form. Of course, you may argue that under some circumstances (maybe when writing a diary or a book) your hypothesis may still hold. However, judging from this study, if it exists at all, this does not seem to be a general effect.
There are some differences in writing style and content, however. For example, introverts seem to use a more concrete and focused language (Gill & Oberlander, 2002). Furthermore, introverts tend to use longer words (Mehl et al., 2006) which may be in line with Alex Stone's comment.
Gill, A., & Oberlander, J. (2002, August). Taking care of the linguistic features of extraversion. In Proceedings of the 24th Annual Conference
of the Cognitive Science Society. Fairfax, VA.
Mehl, M. R., Gosling, S. D., & Pennebaker, J. W. (2006). Personality in its natural habitat: Manifestations and implicit folk theories of personality in daily life. Journal of Personality and Social Psychology, 90(5), 862–877. http://doi.org/10.1037/0022-35126.96.36.1992