C program to find and replace a word in a file
Small suggestions:


Be consistent. What jumps out right away is that you say "This program will perform a find and replace on a single word in a file" but then  your program prompts the user to either write to file or replace a word.  That confuses me as a user of your program as to what it's supposed to do.
Removed. Refer to Difference between int and char in getchar/fgetc and putchar/fputc?

You've declared int c in main(). Why int and not char ?  This also comes up when your case statement reaches the default part. When user enters wrong command, say g, it displays:

Enter 'w' to write to the file, 'r' to replace a word
g
Invalid command:103
Invalid command:10


 So avoid confusing the user and the people who are going to read your code. 
In fgets(old_word, sizeof(old_word), stdin); why do you use sizeof(old_word) ? You already have a constant for WORD_SIZE defined, so there's no need to figure out size of your string.
Consider moving the code from case 'r' for getting user's input to replace_word() function or separate function get_user_input().  What was nice about your case statement is that you started out with case 'w' where you executed a specific function for that case. That's good strategy. Use that case statement to direct the program to specific functions that do specific job.
When user selects to replace a word, you give the user instructions to enter old and new string. That's good - user is not left hanging and is guided in what to do. But when user selects to write to file, there's no instruction and user is just dropped into typing stuff. 

I only clicked Ctrl+D ( to send EOF character since I'm Linux user, not sure how would that work on Windows) because I know that this is what program expects to receive to indicate that there's no more input, but what about a random user who's absolutely unfamiliar with your program and never read your source code ?
On the same related note, how does user exit your program ? In your case statement you only have two cases for specific action, there's no quit option, and neither does it explain to the user that they could use Ctrl+C or Ctrl+D to quit. 
Give the user Enter 'w' to write to the file, 'r' to replace a word  each time an action has been complete. Take for example the example I showed in my first point. User enters wrong command, they see an error and . . . what's next ? The program just sits there waiting for input without explaining to the user what it wants.