If anything happens to you, please let us know right away.
Literal
By-chance your body [on-に] something [subject-が] occur-[conditional-ば], immediately us [to-に] inform-[polite request-てください].
身に何か起こる ('something happens on one's body/self') is an idiomatic way to say 'something happens to someone' — 身 ('body, self') marks the person whose body or life is affected. A more intimate and concerned phrasing than plain '起こったら.'