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The capacity of caching networks has received considerable attention in the past few years. A particularly studied setting is the shared link caching network, in which a single source with access to a file library communicates with multiple users, each having the capability to store segments (packets) of the library files, over a shared multicast link. Each user requests one file from the library according to a common demand distribution and the server sends a coded multicast message to satisfy all users at once. The problem consists of finding the smallest possible average codeword length to satisfy such requests. In this paper, we consider the generalization to the case where each user places L ≥ 1 independent requests according to the same common demand distribution. We propose an achievable scheme based on random vector (packetized) caching placement and multiple groupcast index coding, shown to be order-optimal in the asymptotic regime in which the number of packets per file B goes to infinity. We then show that the scalar (B = 1) version of the proposed scheme can still preserve order-optimality when the number of per-user requests L is large enough. Our results provide the first order-optimal characterization of the shared link caching network with multiple random requests, revealing the key effects of L on the performance of caching-aided coded multicast schemes.