Functionality of vitamin A, Bx, C, D, E, K as used in cosmetics for the skin?
Vitamin A: One study conducted in the 60's claims that vitamin A acid prevents formation of comedones in Acne Vulgaris [1]. Another one conducted in the 70's found no significant improvement of plantar warts when vitamin A acid was applied topically [2].

Vitamin D: Agents like calcipotriene associated with topical steroids are effective against psoriazis [3]. They inhibit keratinocyte proliferation, normalize differentiation and modulate the activity of immune cells [4]. Active form of vitamin D is also effective against vitiligo [5].

Vitamin E: It reduces skin damage induced by UV [6, 8], but it shouldn't be applied on scars [7].

Vitamin K: It prevents bruising after laser treatment and reduces the severity of eventual bruising [9, 10].

Vitamin B12: It is an effective treatment for atopic dermatitis [11] and children with eczema [12]. It could also potentially improve olfactory function when associated with corticosteroids [13].

Vitamin B1: It doesn't protect you from mosquitoes [14].

Vitamin B3: It improves fine lines/wrinkles, hyperpigmentation spots, texture, red blotchiness and sallowness [15, 16].

Vitamin C: It improves wrinkling by new collagen formation [17]. It is useful in sun-induced skin ageing [18, 19] where it increases density of dermal papillae [20].

And here are the results of a study [21] which assessed multiple vitamins:


  Vitamins A, C, E, and B3 have been shown to have potent antioxidant and anti-inflammatory properties, but to achieve optimal effectiveness, products must be delivered in appropriate formulations. Products containing alpha-tocopherol (vitamin E), L-ascorbic acid (vitamin C), retinol (vitamin A), and niacinamide (vitamin B3), are effective for the treatment of photoaging. These compounds have also shown effectiveness in the treatment of inflammatory dermatoses, acne, and pigmentation disorders and wound healing. 


The same study [21] also concludes that combinations are better:


  There is emerging evidence that combinations of vitamins have additive effects that provide enhanced efficacy compared with individual compounds.




References:


Kligman AM, Fulton JE, Jr., Plewig G. Topical Vitamin A Acid in Acne Vulgaris. Arch Dermatol. 1969;99(4):469-476. doi:10.1001/archderm.1969.01610220097017.
de Bersaques J. Vitamin A acid in the topic treament of plantar warts. Dermatologica. 1976;150(6):369-71. PubMed PMID: 1201813. 
Devaux S, Castela A, Archier E, Gallini A, Joly P, Misery L, Aractingi S, Aubin F, Bachelez H, Cribier B, Jullien D, Le Maître M, Richard MA, Ortonne JP, Paul C. Topical vitamin D analogues alone or in association with topical steroids for psoriasis: a systematic review. J Eur Acad Dermatol Venereol. 2012 May;26 Suppl 3:52-60. doi: 10.1111/j.1468-3083.2012.04524.x. PubMed PMID: 22512681. 
Tanghetti EA. The role of topical vitamin D modulators in psoriasis therapy. J Drugs Dermatol. 2009 Aug;8(8 Suppl):s4-8. PubMed PMID: 19702030. 
Gorman S, Judge MA, Hart PH. Immune-modifying properties of topical vitamin D: Focus on dendritic cells and T cells. J. Steroid Biochem. Mol. Biol. 2010 Jul;121(1-2):247-9. doi: 10.1016/j.jsbmb.2010.02.034. PubMed PMID: 20211255.
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Baumann LS, Spencer J. The effects of topical vitamin E on the cosmetic appearance of scars. Dermatol Surg. 1999 Apr;25(4):311-5. PubMed PMID: 10417589. 
Krol ES, Kramer-Stickland KA, Liebler DC. Photoprotective actions of topically applied vitamin E. Drug Metab. Rev. 2001;32(3-4):413-20. doi: 10.1081/DMR-100102343. PubMed PMID: 11139138. 
Shah NS, Lazarus MC, Bugdodel R, Hsia SL, He J, Duncan R, Baumann L. The effects of topical vitamin K on bruising after laser treatment. J. Am. Acad. Dermatol. 2002 Aug;47(2):241-4. PubMed PMID: 12140470. 
Cohen JL, Bhatia AC. The role of topical vitamin K oxide gel in the resolution of postprocedural purpura. J Drugs Dermatol. 2009 Nov;8(11):1020-4. PubMed PMID: 19894369. 
Stücker M, Pieck C, Stoerb C, Niedner R, Hartung J, Altmeyer P. Topical vitamin B12--a new therapeutic approach in atopic dermatitis-evaluation of efficacy and tolerability in a randomized placebo-controlled multicentre clinical trial. Br. J. Dermatol. 2004 May;150(5):977-83. doi: 10.1111/j.1365-2133.2004.05866.x. PubMed PMID: 15149512. 
Januchowski R. Evaluation of topical vitamin B(12) for the treatment of childhood eczema. J Altern Complement Med. 2009 Apr;15(4):387-9. doi: 10.1089/acm.2008.0497. PubMed PMID: 19368512. 
Heilmann S, Just T, Göktas O, Hauswald B, Hüttenbrink KB, Hummel T. [Effects of systemic or topical administration of corticosteroids and vitamin B in patients with olfactory loss]. Laryngorhinootologie. 2004 Nov;83(11):729-34. doi: 10.1055/s-2004-825676. PubMed PMID: 15538662. 
Holzer RB. [Protection against biting mosquitoes]. Ther Umsch. 2001 Jun;58(6):341-6. PubMed PMID: 11441693. 
Bissett DL, Miyamoto K, Sun P, Li J, Berge CA. Topical niacinamide reduces yellowing, wrinkling, red blotchiness, and hyperpigmented spots in aging facial skin. Int J Cosmet Sci. 2004 Oct;26(5):231-8. doi: 10.1111/j.1467-2494.2004.00228.x. PubMed PMID: 18492135.
Bissett DL, Oblong JE, Berge CA. Niacinamide: A B vitamin that improves aging facial skin appearance. Dermatol Surg. 2005 Jul;31(7 Pt 2):860-5; discussion 865. PubMed PMID: 16029679.  
Fitzpatrick RE, Rostan EF. Double-blind, half-face study comparing topical vitamin C and vehicle for rejuvenation of photodamage. Dermatol Surg. 2002 Mar;28(3):231-6. PubMed PMID: 11896774. 
Humbert PG, Haftek M, Creidi P, Lapière C, Nusgens B, Richard A, Schmitt D, Rougier A, Zahouani H. Topical ascorbic acid on photoaged skin. Clinical, topographical and ultrastructural evaluation: double-blind study vs. placebo. Exp. Dermatol. 2003 Jun;12(3):237-44. PubMed PMID: 12823436. 
Farris PK. Topical vitamin C: a useful agent for treating photoaging and other dermatologic conditions. Dermatol Surg. 2005 Jul;31(7 Pt 2):814-7; discussion 818. PubMed PMID: 16029672. 
Sauermann K, Jaspers S, Koop U, Wenck H. Topically applied vitamin C increases the density of dermal papillae in aged human skin. BMC Dermatol. 2004 Sep 29;4(1):13. doi: 10.1186/1471-5945-4-13. PubMed PMID: 15456516. 
Burgess C. Topical vitamins. J Drugs Dermatol. 2008 Jul;7(7 Suppl):s2-6. PubMed PMID: 18681152.