Well, a vaccine is a rather vague and generic term. Are you talking about an FDA approved product? An experimental vaccine? A vaccine whose use is contra-indicated for certain people? etc. etc.

If I may - perhaps one alternative question is "Can an unapproved vaccine product cause harm?". With that re-wording, the answers become a bit more clear. Yes, chemicals used in the formulation maybe toxic, or the final vaccine might give rise to an auto-immune response, or could induce anaphylaxis in certain people. There could be cell debris/endotoxins that a purification process doesn't remove properly and causes problems. Maybe the vaccine contains an attenuated pathogen that can "revert", etc, etc.