Comment on Who counts as “Amish”? by Erik/Amish America.
Wayne the short answer is the deacon is the one in charge of organizing moneys when there is a need in the community. More on the deacon and other leaders’ roles here: http://amishamerica.com/who-leads-the-amish-church/
Amish believe strongly in mutual aid, supporting one another when there is hardship, catastrophe, high medical bills, etc. Beyond this, Amish also contribute financially and with elbow grease to various charities that reach beyond their immediate community.
I have never experienced a formal pass-the-hat collection in an Amish church service. It is handled outside of the service. The one place I have encountered a formal collection is at New Order Amish Sunday School.