Are graphic designers required to print things for clients?

Required?…. no.

Can you?…. if you want to.

You can either subcontract printing services and then mark up the costs and pass them on to your clients. This makes you responsible for paying all printing costs regardless of whether your client pays you or not. You can offset the risk by requiring deposits on printing services. This means you can profit a bit though pricing markups.

Or, you can merely deliver press-ready files and let the client find a print provider and handle all reproduction. This removes any risk of you carrying costs, but may forgo some revenue from marking up the printing fees.

Both business models work.