Final answer:
The connectivity issue where a workstation receives an APIPA address and cannot reach the VLAN gateway is most likely caused by the switchport being mistakenly configured for 802.1q trunking.
Step-by-step explanation:
If a workstation is connected to the network and receives an APIPA address but cannot reach the VLAN gateway, while other PCs in the VLAN subnet can communicate with the VLAN gateway and access websites on the Internet, the most likely source of the connectivity problem is that the switchport is configured for 802.1q trunking. APIPA addresses are automatically assigned when a device cannot contact a DHCP server to obtain an IP address, which suggests a connectivity issue between the workstation and the VLAN's network infrastructure. Workstations that are supposed to be on an access port will not be able to communicate properly if the port is configured for trunking, as they won't be assigned to the correct VLAN.