Die Übersetzung "Er macht zwei Bestellungen." klingt in deutschen Ohren schrecklich und plump. Richtige Übersetzungen für "He makes two orders" sind beispielsweise: "Er gibt zwei Bestellungen auf.", "Er bestellt zwei Mal."
Die Übersetzung "Er macht zwei Bestellungen." kann im Deutschen auch als "Er bearbeitet zwei Bestellungen." verstanden werden. Dies würde im Englischen vermutlich mit "He processes two orders." zu übersetzen sein. "Martin macht gerade die zwei Bestellungen, die gestern Abend noch herein gekommen sind und ich mache die Reklamation von heute früh." so könnte sich ein kurzer informeller und mündlicher Statusbericht an den Chef anhören.
It's neither very clear, nor a good example. More common would be 'Er gibt zwei Bestellungen auf' (places two orders). Less common and depending on the context would be your 'fulfilling'. If he was working at a warehouse, he might handle or complete two orders. 'to place' is more likely anyway. Fun fact: Searching for 'Bestellung machen' on Google, each hit on the first page links to a dictionary, not actual usage. I wouldn't consider it decent German.
I am an native English speaker and "make" makes much more sense to me, I'd only say place if there was going to be a long time before the order was delivered. I would never place an order in a restaurant. If I placed an order, I'd expect to return another day to receive it (eg. I placed an order for a new car).
This is one of those times when language isn't entirely logical or intuitive.
The most common phrasing, in the US, is to "place" an order. I'm sure of that, because in my company we receive orders from all over the US by phone, and what people almost always say is "I'd like to place an order."
"He does two orders" or "he makes two orders" would not be understood.
[Native US English speaker]
Apparently there are regional differences, but "order" or "place an order" are the only forms I've encountered myself in this context - and I've lived in various parts of the country and do business across the country.
Apparently there is a legal context for "make an order," but that involves a court order, rather than a commercial transaction.
Here's a page that might be useful: https://forum.unilang.org/viewtopic.php?t=34208
@Soglio: "Makes two orders" sounds fine to me (also native English speaker). I tried a Google Ngram search, and "make an order" appears to be more common than "place an order" (though "place orders" is more common than "make orders").
"Make an order" sounds a bit more informal to me, which may explain why you wouldn't hear it from customers over the phone.