So I don't know if anyone found the fixes they needed in order to make the game work but here is what helped me.
First try changing LoR to run as admin:
Right click the shortcut for Legends of Runeterra, click properties.
In the compatibility tab check "run this program as administrator", hit OK.
Next uninstall all 3rd party antivirus programs:
Hit the Windows key and type in "add or remove programs" then search for your antivirus and right click to uninstall.
Restart your computer for these changes to take effect.
Next send a screenshot to confirm your firewall isn't blocking Legends of Runeterra:
Hit the windows key and type "firewall". Click "check firewall status" and hit advanced on the left.
Click inbound rules, then click "new rule" in the top right.
For rule type select program.
For program path click browse and locate C:\Riot Games\LoR\live\Game\LoR.exe
Click next to allow the connection.
Make sure all 3 profiles are checked, then press next. Repeat this process for C:\Riot Games\Riot Client\RiotClientServices.exe
Send a screenshot of the 2 exceptions you made to confirm they are correct.
If this still isn't working, reinstall Legends of Runeterra completely:
Uninstall LoR from the "Add or remove programs" menu.
Open file explorer (windows + E)
Delete the folders C:\Riot Games\LoR and AppData\Local\Riotgames
Re-download and install LoR as administrator (right click installer, run as admin)
I got great help from the support team, though it did take awhile for them to get back to me. What seemed to do the trick was uninstalling my antivirus (in my case it was Malware-bytes) and making sure that my firewall was not blocking the program. I then uninstalled the game completely (making sure nothing of the game or the launcher was anywhere on my computer.) then reinstalling it. After those four steps above it worked if, if this doesn't work for you I would try to just send in as many tickets as need be til you have it working.
Good luck and have fun.