Research results are somewhat inconsistent but generally seem to be unfavorable to vitamins alone (in contrast to vitamins in combination with other things).
From "Prevention and treatment of the common cold: making sense of the evidence" (2014):
Zinc appears to be effective in reducing the number of colds per year, at least in children. (...) vitamin D and echinacea showed no evidence of benefit. Vitamin C may provide some benefit in people under physical stress (e.g., marathon runners or soldiers in subarctic environments), but no meaningful benefit has been shown for the average patient. (...) Evidence for interventions aimed at preventing and treating the common cold is frequently of poor quality, and results are inconsistent. The best evidence for the prevention of the common cold supports physical interventions (e.g., handwashing) and possibly the use of zinc supplements.
(Caution: be aware, that intranasal zinc may induce some adverse effects, i.e. anosmia syndrome.)
From "Vitamin C for preventing and treating the common cold." (2013):
The failure of vitamin C supplementation to reduce the incidence of colds in the general population indicates that routine vitamin C supplementation is not justified
From "The effect of exercise on prevention of the common cold: a meta-analysis of randomized controlled trial studies." (2015):
Dietary supplements, such as vitamin C and E, are used by many people, especially athletes. The users often believe that high dosages of supplements improve health (resistance to illness and disease) and physical performance. These assumptions are, however, generally not supported in the scientific literature.
From "Vitamin D for prevention of respiratory tract infections: A systematic review and meta-analysis" (2012):
On the basis of this study, we can conclude that vitamin D is useful in prevention of respiratory tract infections. But looking at the availability of only five clinical trials there is need of conduction of more clinical trials so that more valid conclusion can be reached.
From "Effect of vitamin D3 supplementation on upper respiratory tract infections in healthy adults: the VIDARIS randomized controlled trial." (2012):
(...) monthly administration of 100,000 IU of vitamin D did not reduce the incidence or severity of URTIs in healthy adults.