Writing a reusable method in which only some of its parameters are different
What you're looking for is an Optional Argument. When using optional arguments, you have to supply a default value. So, you just have to check for that default value, and switch your control flow appropriately. It would look something like this. (Note I used empty quotes because String.Empty isn't a constant expression.)

private string FindFullRowForProvider(string id, string lastName, string firstName, string devSystem = "")
{
    //maybe some code here

    if (devSystem == String.Empty)
    {
        //handle that case
    }

    //maybe some more code here

}


But... this is Code Review after all, so let's take a look at what you've done. 




private string FindFullRowForProvider(string id, string lastName, string firstName, string devSystem)



I'm making an assumption that id is tied to lastName and firstName,  but it looks like you're missing a class in your design. Creating a Person class would simplify the signature an awful lot. Be wary of passing lots of strings to a method. 

private string FindFullRowForProvider(Person somePerson, string devSystem = "")




I like your use of String.Empty. I really do prefer it over empty quotes. I think it makes for nice readable code. Well done.


string fullRow = string.Empty;





Maybe I'm missing something, but I think declaring a character array for delimiter is overkill here.


char[] delimiter = new char[] { '|' };
int columnLocation = 0;

string[] columnheaders = streamreader.ReadLine().Split(delimiter);



I'm not very good at C#, so there may be an even simpler way, but you could certainly just split it like this.

string[] columnheaders = streamreader.ReadLine().Split(new char[] { '|' });




Nice use of brackets and formatting here. 


        if (npi.Trim().ToUpper() == _NPI.Trim().ToUpper()
            &amp;&amp; _LName.Trim().ToUpper() == lastName.Trim().ToUpper()
            &amp;&amp; _FName.Trim().ToUpper() == firstName.Trim().ToUpper()
            &amp;&amp; _dev.Trim().ToUpper() == devSystem.Trim().ToUpper())
        {
            // woot! we found the row of this guy.
            return fullRow;
        }