I arranged a meeting today with Marlene Friedman, Foster Care Supervisor; Phyllis Manners, Family Preservation Supervisor; Tammie and me to talk about scheduling to avoid overtime.

Should me or I be used after Tammie?

Me - definitely me.

You don't say with I, do you?

This is good use of semicolons in a list. However, once you start with semicolons, you have to keep using them. Your list should be:

I arranged a meeting today with Marlene Friedman, Foster Care Supervisor; Phyllis Manners, Family Preservation Supervisor; Tammie; and me to talk about scheduling to avoid overtime.