Some definitions appear to be in order:
machine = the physical / virtual server on which an instance of SQL Server is installed.
SQL Server = software that runs on a machine. There can be one or more instances of SQL Server installed on a machine.
For what you're trying to accomplish, you need permissions to
1) connect to the machines running the SQL Server services
2) view the status of services running on those machines.
If you don't have 1), it's all over. Even if you do have 1), you may not have 2): that will be up to your sysadmin.
At no point do you need to login to any SQL Server: this has nothing to do with SQL Authentication whatsoever. You can have an SQL Login and DB access on a SQL Server, but without the rights to connect to the machine running that SQL Server, well, you won't get in. Also, even if you are allowed to connect to the machine, if the SQL Server service is not started, your SQL Authentication will fail.
You'll need to use the Get-WmiObject instead, it allows you to pass credentials. You can look it up.
Get-WmiObject win32_service -comp "ServerName" | select __server,name,startmode,state,status