In order to configure the built-in NIC on a Lenovo ThinkPad T440p to use specific VLANs, you need to install the newer drivers that include Advanced Networking Services (ANS) VLANs. The full package that is required is the Intel® PROSet and Advanced Network Services* (ANS).
The built-in NIC on a Lenovo ThinkPad T440p is the Intel® Ethernet Connection I217-LM, go to this link and select your operating system to download the correct drivers. My T440p runs Windows 7, I downloaded the x64 drivers from here and after installing them I could see the VLANs tab (amongst others) in the NIC properties that weren’t there previously:
As you can see above, I configured the NIC to use VLAN ID 3. Once done, I was able to plug my laptop into a switch and access the network successfully.