No, Invoke-WebRequest is present in version 5 (I am running 5.1 and it is still there). I found a post where it is explained, that the alias wget is deprecated in 5.1, but the original function is still there.
The only time where Invoke-WebRequest would not be present for you would be if you ran version 2 or lower. This is only the case on Windows 7. You can check this if you run a powershell window (Just type powershell into your windows search or press the Windows Key + R, type powershell and run it).
There type the command:
$PSVersionTable.PSVersion
If this shows for the Major column a version lower than 3 you are running on an old version and a solution for you would be to update your version here then it should run.
As much as I want to fix this game I really don't want to find workarounds for web requests and Json parsing in version 2.0. I did this at one point for other stuff and it is a pain in the arse :P . (Also the version 2 on my system is somehow busted and I would either need to reinstall windows, setup a windows 7 VM or something similar just to test stuff.)