enters a custom approval workflow before it shows up on the calendar.
This is fine. Approval workflows are out-of-the-box, and you don't need even SharePoint Designer.
A manager has requested that an email be sent at the beginning of each week which shows upcoming absences for the week.
This kind of requests cannot be fulfilled with workflows I think for a ton of reasons. SharePoint workflow are item-event-driven, information is item-scoped and cycling is not easy life in terms of design and performance.
For such a requirement I'd try another approach.
I'd first implement a powershell script which gathers and formats the required information from the list and sends it to the recipients.
Then I would schedule the script through the Task Scheduler to start at the beginning of each week.
As per the comments "It sounds like you have a solution in mind: would you mind posting it as an answer and I can pass it along to the group who handles site-deployment requests?"
First, you should ask your farm admin, whether Powershell Remoting is enabled and the parameters you require to connect to a powershell session remotely. They should also make sure that you are assigned to the required permissions to do your tasks.
Second you just need to install a Powershell and then add the sharepoint snapin. This also should be advised by the farm admin.
Third just start and try scripting what you want. To send emails from powershell you'll probably need some information from the SMTP server admin. To know how to access sharepoint data read technet sharepoint powershell docs.