Hi Jaime, you can use 発注 and 注文 as same words in general conversation. As far as I know, 発注 is often used in case that companies or organizations buy something (e.g. road construction, huge machines, a lot of pens or lunch box for employees), on the other hand, 注文 is used in case that people buy something privately.
So using 注文 is much more natural than 発注 in daily conversation. I also do not use 発注 in my daily life, but often use it at work. Thanks.