There are different opinions on this topic. Literature does not provide clear evidence for Vit D supplements. Although it seems like an interesting research topic. I am sure it will provide interesting and helpful information. I have seen interesting findings related to root resorption, but is anecdotal, no clear evidence found in the literature.
The modern understanding of vitamin D goes well beyond its functions in calcium metabolism. Every human cell has 48 nuclear receptors that up- and down-regulate genes.
One of these receptors (called the Vitamin D receptor or VDR) responds to at least three vitamin D compounds, each of which, with some overlap, regulates different genes.
http://www.ncbi.nlm.nih.gov/pubmed/24116037
There is evidence from epidemiology, microbiology, and randomized controlled trial that vitamin D status and gum disease are closely related, although not by the mechanism you suggest. See these three articles:
vitamin D is critical for dental health. A number of randomised trials have shown that it helps prevent dental caries although these trials are very old
Hujoel PP. Vitamin D and dental caries in controlled clinical trials: systematic review and meta-analysis. Nutr Rev. 2013 Feb;71(2):88-97. doi: 10.1111/j.1753-4887.2012.00544.x. Epub 2012 Nov 9.
There is also a randomised clinical trial showing it helps gingivitis.
Hiremath VP, Rao CB, Naik V, Prasad KV. Anti-inflammatory effect of vitamin D on gingivitis: a dose-response randomised control trial. Oral Health Prev Dent. 2013;11(1):61-9. doi: 10.3290/j.ohpd.a29377.
Vitamin D3 stimulates tooth movement, and dietary calcium seems to reduce it.
1. Bartzela T, Türp JC, Motschall E, Maltha JC. Medication effects on the rate of orthodontic tooth movement: a systematic literature review. Am J Orthod Dentofacial Orthop. 2009 Jan;135(1):16–26.
All active form of vitamin D3, 1,25-dihydroxycholecal- ciferol, reportedly increased the number of osteo- clasts.
1. Killiany DM. Root resorption caused by orthodontic treatment: an evidence-based review of literature. Semin Orthod. 1999 Jun;5(2):128–33.