from Wikipedia, so read it with a grain of salt:

Earliest uses

Slate's Daniel Engber dates the modern usage to 1974 college campuses, where it was perhaps an outgrowth of the Vietnam War.^[5]

The origins of the phrase are uncertain, with some speculating that
it may refer to being "out in the open" or "ready for action."
According to Engber many soldiers do go without underwear to "increase
ventilation and reduce moisture." The earliest known use of the term in
print is January 22, 1985 when Jim Spencer wrote in the Chicago Tribune:

<blockquote>

"Furthermore, colored briefs are 'sleazy' and going without
underwear ('going commando', as they say on campus) is simply gross".

</blockquote>

The origin of the term "going commando" may connect to the tradition
in military units of Scottish heritage to not wear underwear under
their kilts. The first Commando units were raised and trained in World War II
in Scotland (which is where the present-day Commando Monument is). It
is said that Scottish soldiers once used the expression "regimental,"
to describe the state of wearing no shorts under a kilt; allegedly,
Scottish drill instructors carried a long stick with a mirror on the
end to verify that soldiers were properly regimental.^{[citation needed]}

In the British Forces, the term 'going commando' actually refers to
not washing and generally being dirty, as Royal Marine Commandos and
other elite regiments tend to have a reputation of doing when on
exercise, or on patrol.^{[citation needed]}