You may be out of luck here as the 5.1 C# client is definitely not compatible with Windows 10. You can try some of the C++ redistributable libraries to see if this may be fixable with those, but ultimately you might need to use an earlier version of Windows if you absolutely must connect to vSphere/ESXi 5.1 (which is also out of support now).
I think you may be correct. I know that this is not a Microsoft forum though I'm hoping that someone might have run across this. I tried to add in the windows feature ".NET Framework 3.5 (includes .NET 2.0 and 3.0). The problem is that after I click the checkbox and reboot, the checkbox is still empty.
I'm guessing group policy though I'm hoping that someone might have run across this in the past and could point me in the right direction.
Thanks for the assistance.
There is a way to get it to work.
Simply open the "Turn Windows features on or off" -> Check ".NET Framework 3.5" then restart after install.
You should be able to install the vsphere client 5.1 on windows 10.
Thank you! Installing ".NET Framework 3.5" (no reboot required for me) allowed the install to continue for me on a clean updated Windows 10 install Jan 19, 2020.