Actually, the word does show up in dictionaries, even if it is considered “nonstandard.” Here’s part of what Merriam-Webster wrote about why it deems “irregardless” an actual word:
It has come to our attention lately that there is a small and polite group of people who are not overly fond of the word irregardless. This group, who we might refer to as the disirregardlessers, makes their displeasure with this word known by calmly and rationally explaining their position … oh, who are we kidding … the disirregardlessers make themselves known by writing angry letters to us for defining it, and by taking to social media to let us know that “IRREGARDLESS IS NOT A REAL WORD” and “you sound stupid when you say that.”We define irregardless, even though this act hurts the feelings of many. Why would a dictionary do such a thing? Do we enjoy causing pain? Have we abdicated our role as arbiter of all that is good and pure in the English language? These are all excellent questions (well, these are all questions), and you might ask them of some of these other fine dictionaries, all of whom also appear to enjoy causing pain through the defining of tawdry words.--Irregardless: Regardless— The American Heritage Dictionary of the English Language, Fifth Edition, 2018--Irregardless: In nonstandard or humorous use: regardless.— The Oxford English Dictionary, 2nd edition, 1976--Irregardless: without attention to, or despite the conditions or situation; regardless— Cambridge Dictionary (dictionary.cambridge.org), 2018The reason we, and these dictionaries above, define irregardless is very simple: it meets our criteria for inclusion. This word has been used by a large number of people (millions) for a long time (over two hundred years) with a specific and identifiable meaning (“regardless”). The fact that it is unnecessary, as there is already a word in English with the same meaning (regardless) is not terribly important; it is not a dictionary’s job to assess whether a word is necessary before defining it. The fact that the word is generally viewed as nonstandard, or as illustrative of poor education, is likewise not important; dictionaries define the breadth of the language, and not simply the elegant parts at the top.We must confess that of the charges leveled against irregardless, the one asserting that it is not actually a word puzzles us most. If irregardless is not a word, then what is it, and why is it exciting so many people who care about words? Of course it is a word. You may, if you like, refer to it as a bad word, a silly word, a word you don’t like, or by any one of a number of other descriptors, but to deny that a specific collection of letters used by many people for hundreds of years to mean a definite thing is a word is to deny the obvious.
You can read the rest of the dictionary explanation here.