In this paper, we first introduce the notion of a (mild) $C$-existence family in complete random normed modules, then we prove that a (mild) $C$-existence family can guarantee the existence of the (mild) solutions of the associated abstract Cauchy problem in the random setting. Second, we investigate several important properties peculiar to locally almost surely bounded $C$-semigroups in complete random normed modules, which are not involved in the classical theory of $C$-semigroups. Finally, based on the above work, some relations among $C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules are established, which extend and improve some known results. Besides, an application to a type of stochastic differential equations is also given.