A number of existing Access Web Apps are reporting an error "403 Forbidden" when you attempt to view them from an Internet browser and "Server was unable to process request —> Only users who can View Pages can list Apps." when used from the Access client.
This issue was resolved Saturday, June 7th at approximately 11:00PM UTC.
Details from the Office 365 Service Status:
Closure Summary: On Monday, June 2, 2014 at approximately 11:00 PM UTC, engineers identified an issue in which some customers were unable to open SharePoint Online Access Web Apps that were created prior to May 23, 2014. Investigation determined that a database misconfiguration in the capacity which stores Access Web App data caused the issue. Engineers implemented a fix across the affected capacity which remediated impact. The issue was successfully fixed on Saturday, June 7, 2014 at approximately 11:00 PM UTC. A post-incident report will be available on the Service Health Dashboard within five business days.