I'm trying to write a PHP script which will handle sorting of a CSV file by one or multiple columns and outputting the result to another file.
Is there a way to sort the CSV file without loading it entirely into memory?
No there is no reasonable way. You need the data in memory to compare and write in a file.
You could try a bubblesort if you know the length of each line. Read one line of origin and last line of a new "ordered" file. Compare them and append or prepend in new file. After this iteration do again with the new file as origin until it is sorted.
You should use a database like MySQL.