Function that loads a random image using an array of paths to images

To answer your immediate question, you need to initialise your flagImageSources once, and store it as a field in the class. That allows you to split the two actions into separate methods - one to get your list of flags, and one to get a random flag.

public partial class Form1 : Form
{
    private string[] flagImageSources;

    private void InitializeImageSources()
    {
        flagImageSources = new string[196];
        StreamReader flagImageSourceReader = File.OpenText("flagImageSourceFile.csv");

        for (int i = 0; i < flagImageSources.Length; i++)
        {
            string s = flagImageSourceReader.ReadLine();
            flagImageSources[i] = s;
        }
    }

    private string GetRandomImageLocation()
    {
        Random numberGenerator = new Random();
        int imageNumber = numberGenerator.Next(0, 196);

        return flagImageSources[imageNumber];
    }
}

But I think we have some other concerns here.

Firstly, you open your file but forget to close it again. C# has a handy way of making sure this doesn't happen - something that gets opened and closed should implement IDisposable (and StreamReader does just that). So you can change your code a little to make use of that, like this:

        using (StreamReader flagImageSourceReader = File.OpenText("flagImageSourceFile.csv"))
        {
            //do stuff with flagImageSourceReader
        }

Secondly, you're using the magic number of 196 in a couple of places in your code - both to set the initial size of your array, and to set the upper bound of your random number generator. I would suggest that rather than using a fixed-size array, you should use an IList, which is dynamically sized. Then, if North and South Korea should merge, or Scotland or Catalonia or Tibet become independent countries, your code will still work even as you add or remove flags :)

Here is my final code that solves both of those problems:

public partial class Form1 : Form
{
    private IList<string> flagImageSources = new List<string>();

    private int NumberOfFlags => flagImageSources.Count;

    public Form1()
    {
        InitializeComponent();

        InitializeImageSources();
        flagPictureBox.ImageLocation = GetRandomImageLocation();
    }

    private void InitializeImageSources()
    {
        using (StreamReader flagImageSourceReader = File.OpenText("flagImageSourceFile.csv"))
        {
            while (!flagImageSourceReader.EndOfStream)
            {
                flagImageSources.Add(flagImageSourceReader.ReadLine());
            }
        }
    }

    private string GetRandomImageLocation()
    {
        Random numberGenerator = new Random();
        int imageNumber = numberGenerator.Next(0, NumberOfFlags);

        return flagImageSources[imageNumber];
    }

    private void submitBtn_Click(object sender, EventArgs e)
    {
        flagPictureBox.ImageLocation = GetRandomImageLocation();
    }
}