In other words, what DML is going to happen that will fire the trigger to update table orders that will update the column item_reorder? One trigger, does not "fire another", a trigger does some action, based on an action that has occurred in the database. That triggered action could fit the definition of some other trigger, that causes the second trigger to "fire".
I think what you are after is more like a single trigger. You have a process that will update the table orders and update the column item_reorder to yes, then you just need a trigger that will fire after that update to insert the appropriate data to the reorder table.
Dave