Vitamin C does NOT cause constipation.
More Information: Not only does vitamin C not cause constipation, the side effects of too much vitamin C have the exact opposite effect. For example, vitamin C supplements are known to have a diuretic effect helping the body eliminate excess fluids. At high doses, vitamin C supplements can cause diarrhea. 
Does Vitamin C Treat Constipation?
Though it may not cause constipation, the National Library of Medicine warns that there is not enough evidence to support using vitamin C supplementation as a treatment for constipation. 
Study Reveals Constipated Children Have Lower Vitamin C Intake
Emerging studies are indicating that there is a link between vitamin C and constipation.
A 2008 study published in The Journal of Paedetrics and Child Health, evaluated the diets of 368 children in relation to their constipation and discovered that besides being deficient in fiber, which was expected, those suffering from constipation were also deficient in vitamin C, folate, and magnesium. 
 “Vitamin C (Ascorbic acid).” University of Maryland Medical Center | Home. N.p., n.d. Web. 1 Mar. 2012. <http://www.umm.edu/altmed/articles/vitamin-c-000339.htm>.
 “Vitamin C (Ascorbic acid): MedlinePlus Supplements.” National Library of Medicine – National Institutes of Health. N.p., n.d. Web. 1 Mar. 2012. http://www.nlm.nih.gov/medlineplus/druginfo/natural/1001.html
 Lee, WT. “Increased prevalence of constipation in pre-school children is attributable to under-consumption of plant foods: A community-based study..” Journal of Paedetrics and Child Health 44.4 (2008): 170-175. Print.