The reason that your field is blank is that you are looking at a different field. You appear to have created a task-level ECF and have populated it and used it to good advantage in task-level views. However–on the Resource Assignments view, you are looking at assignment-level ECFs. The ECF has the same name–but it is a different field.
You can roll down task and resource custom fields to assignment level. On the New Custom Field or the Edit Custom Field page in Project Web Access, click Roll down, unless manually specified in the Calculation for Assignment rows section.
