Can a computer function without one?

This depends on your definition of peripheral devices. A computer certainly does not require I/O devices such as displays, keyboards or printer to function, although most will be designed to interact with the world to some extend.

A traditional CPU requires additional parts, such as memory chips, to function, but micro controllers or more modern system-on-a-chip designs offer more or less complete solutions within a single chip: volatile and non-volatile memory, video, network, and processing cores.

These devices are designed to function without, or with a minimum of low-cost, typically passive, additional electronic parts. Whether you'd consider such a system one with or without peripheral depends on your definition of peripherals though.