Yes, there are lots of people on the forums who have printing businesses
I don't know if what you described would be called "fulfillment", since family reunion shirts and event shirts are usually shipped all directly to the customer.
Fulfillment involves warehousing, individual order processing (one shirt at a time to end customers), things like that.
But I know lots of people who do printing for others to fund their other ventures (like their clothing lines)
You can learn all about it in the different areas of the forum that deal with your printing type.