Can one person be in charge of document management?

Yes! There can be a single document manager dealing with document management efficiently. It usually depends on the influx of work. Sometimes, it can be good when different people help each other out, especially when there isn't enough work in their department. But it is recommended to distribute the work to the experts. A website designer should design a website instead of working in customer support. Hence, give the job of document management to the field expert.