Orifice if from French, from late Latin orificium, from os, or- ‘mouth’ + facere ‘make’. 

Orifice came to English during the late Middle English period (1150-1500), specifically before 1425.  The Oxford English Dictionary cites Guy de Chauliac's "Grande Chirurgie" as an early example of the word's usage, before 1425. 

Examples of orifice used in a sentence: 
"I was stuffing cake into every available orifice." 
"The water is expelled from the branchial chambers by one or two tubes opening by one orifice in most Batrachians." 
"There are likewise two pulmonary veins, entering the left atrium by one orifice." 
"The funnel, which is not large, appears to open, as a rule at least, into the segment in front of that which bears the external orifice." 
"Intromittent organ of male lying within the genital orifice. 
"Before passing into the pupal stage, the larva partially closes the orifice of the tube with silk or pieces of stone loosely spun together and pervious to water."

Just the facts:

ORIFICE Definition & Meaning - Merriam-Webster