A surprise. The possibility is that it’s simply a form of flute, ie to play on the flute, whence it came to mean to mock or deride, which I guess makes sense. If someone is pompously orating and you respond by tooting on your flute you’re clearly not taking them seriously. Here’s OED:
Etymology: First recorded in 16th cent.; possibly special use (preserved in some dialect) of floute , Middle English form of flute v. to play on the flute. Compare a similar development of sense in Dutch fluiten to play the flute, to mock, deride.
BTW I don’t know whether it’s just me but the flout/flaunt error seems to be everywhere these days and crops up in the most unlikely places (NYT for instance). Perhaps it’s owing to the custom now of using online contributors for some pieces rather than established journalists but whatever the reason it annoys the hell out of me.
And for the sake of completeness here’s OED on flaunt, the origin of which is unknown.
Etymology: Of unknown origin.
The monosyllables of similar ending are (except perhaps gaunt) all from French; but no French word is known which could be the source. Possibly the word may be an onomatopoeia formed with a vague recollection of fly, flout and vaunt. Prof. Skeat compares modern Swedish dialect flankt loosely, flutteringly ( < flanka to flutter, waver), also modern German (Bavarian) flandern to flutter, flaunt; but the late appearance of the word in English makes it doubtful whether any connection exists.