A field initializer cannot reference the non static field, method, or property 'NecroBomber.Form1.path'

To solve the compile-time issue, put the initialization in the constructor. You're simply not allowed to refer to one instance variable from the initializer of another. This is laid down in the C# specification in section 10.5.5.2:

A variable initializer for an instance field cannot reference the instance being created. Thus, it is a compile-time error to reference this in a variable initializer, as it is a compile-time error for a variable initializer to reference any instance member through a simple-name. In the example

   class A
   {
       int x = 1;
       int y = x + 1;   // Error, reference to instance member of this
   }

the variable initializer for y results in a compile-time error because it references a member of the instance being created.

It's unclear that you even need path as an instance variable, to be honest:

public class Form1 : Form
{
    private string[] userNames;

    public Form1()
    {
        string path = "http://mywebsite.com/accounts.txt";
        userNames = File.ReadAllLines(path);
    }
}

Or if it's always going to read from that path anyway, you could just inline it:

public class Form1
{
    private string[] userNames = 
        File.ReadAllLines("http://mywebsite.com/accounts.txt");

}

Not that either of these will actually work - File.ReadAllLines takes a local filename - not a URL. You'll need WebClient or something similar instead.

I strongly suggest you take a step back and work out what the logical state of the form is. Loading user names from a web site on construction feels like an odd thing for a form to do, to be honest. Additionally, you shouldn't perform potentially long-running operations (like fetching a web page) on the UI thread.

There's a lot to consider about your code - far more than the immediate compilation problem.