The most formal is to 'order food for delivery'. We might say 'get delivery', 'order food' or 'get take-out' (even though it's being delivered we might still say take-out).
Depending on the region, some restaurants/services make the meaning obvious. For example, instead of:
"I am going to search on the internet for cat pictures", it is common for people to say:
"I am going to Google cat pictures"
In the same way, in England, it is not uncommon to hear:
"We should get Pizza Hut/Chinese/JustEat"
And in these cases it can be obvious that we want the food to be delivered, not that we will go to the restaurant to eat it.