IPv6 addresses do not follow any type of “class” structure like IPv4 addresses did (Class A, B, C) so all IPv6 networks basically use a Variable Length Subnet Mask (VLSM).

Is there an easy way (like via a .Net library) that allows me to get the IP address and subnet mask of the local machine that the application is running on?

You can find your IP address on any Windows computer by entering a specific code in the Command Prompt, or by going to a IP-displaying website.