Here is an Ngram chart of usage frequency for unironically for the period 1900–2019 (the corresponding Ngram graph is here, in case the chart URL no longer works):
As the chart indicates, the word has been in detectable use for more than 100 years and has grown significantly more common during the period since the year 2000 than it was before that.
A Google Books search for unironically finds a small swarm of early occurrences in the period 1921–1923. From Arnold Bennett, "Mr. Prohack," in The Delineator (November 1921):
"You will get used to it, Arthur," said Sir Paul indulgently but not unironically. "You're in a nervous state and your judgment's warped. Now, I never even heard your famous clock strike ten."
From Edmund Blunden, The Bonadventure: A Random Journal of an Atlantic Holiday (1923):
Bicker, the editor, instead of reviewing his admired literature in his journal, lengthened breakfast by doing so there viva voce. He was all for Bœotian situations, and, on occasion, his cold re-dishing was tactfully ended by a relief conversation on religion, the keynote of which was in the unironically meant remark: "He was darned religious, but he was a darned good man."
And from Richard Curle, Into the East: Notes on Burma and Malaya (1923):
The town [Singapore], like a camel in the desert, was living on its hump, was living on hope. Its huge fabric reminded one unironically of a whited sepulchre; it is the product of trade and without trade it is a mere shell. Well, of course, not altogether, because its geographical position is its ultimate value, but to a large extent. Singapore was urgently in need of the reviving breath of industry.
The sense of the term is in each case, simply, "not ironically"—that is, without any awareness of or intention to express a contrast or incongruousness between a thing said and the thing or situation about which it was said.
Bennett and Blunden were Englishmen and Curle was a Scot, so it may be that the word first came into literary use in Great Britain. Be that as it may, the rise of unironically did not begin in earnest until the mid-1970s, after which it seems to have made considerable progress into mainstream written usage.
Even so, it bears noticing that a Google Ngram chart mapping the frequency of unironically (blue line) against the frequency of ironically (red line) for the period 1900–2019 suggests that the latter has made far more progress in frequency of usage over the past half-century than the former has (again, the corresponding Ngram graph is here):
In any event, unironically is certainly a word in good standing (outside the precincts of Microsoft Word, anyway), with a clear meaning and consistent usage over many decades to support it. Who (besides Microsoft) could ask for anything more?