We are using the Shared Files feature for functionality similar to this. The basic steps are:

• We set up Shared Files instances configured to be accessed only by each department group and the administrative coordinators
• Before each payroll period, administrative coordinators add an empty "timecard" Excel spreadsheet to the Shared Files instances for each user
• Users download and fill out their own timecard, then upload when complete
• At the end of the payroll period, admin coordinators download all timecards and use them to do time entry in our payroll system

It's missing niceties like workflow approvals, etc., but it seems to be working well for the groups that are using it.