You can't open the UI as a regular user. However, there is a way to enable Wireguard to work with a regular user in Windows 10, giving the user the ability to freely start and stop the Wireguard tunnel.
Since Wireguard runs as a service in Windows, you can change the permissions for that service, without having to give the user more privilege than it needs to have.
Configure the WiregaurdWireguard tunnel on the machine using an admin account and the GUI then start the tunnel
Open a command prompt and change the permissions for the service
sc.exe sdset WireGuardTunnel$NameOfTheTunnel "D:AR(A;;CCDCLCSWRPWPDTLOCRSDRCWDWO;;;SY)(A;;CCDCLCSWRPWPDTLOCRSDRCWDWO;;;BA)(A;;CCLCSWRPWPDTLOCRRC;;;WD)(A;;CCLCSWLOCRRC;;;IU)S:(AU;FA;CCDCLCSWRPWPDTLOCRSDRCWDWO;;;WD)"
You can then start and stop the tunnel using those commands. Don't use the GUI to start/stop at this point
sc stop WireGuardTunnel$NameOfTheTunnel
sc start WireGuardTunnel$NameOfTheTunnel
Create a batch file, one for each command, like WireguardON.bat and WiregaurdOFF.bat and put them somewhere the client could access (on its desktop or something)
Logout and log back in with the user account. It should be able to start and stop the service, even after a reboot.