'The Army' refers to the British army who make up part of the armed forces of the United Kingdom. Mr.Smith could be associated with it as he has the title of 'Major' however has he arrives in a suit it could mean he is part of a higher organization.
They first appear at the very start of The Demon Headmaster Takes Over, when they are assigned to reopen and collect evidence from the Biogenetic Research Centre after six months after the events of The Demon Headmaster Strikes Again.
After arriving they are met by Mr.Smith who explains the situation, they must restore power to the centre and not enter until he returns from inside the building. This is probably because anything inside was top secret and the government didnt want such infomation getting into the public eye.
The Army are later seen with flamethrowers ready to burn any remains of the now dead Creeper, it can be presumed that all cases of the Creeper were destroyed, the centre was cleaned up with evidence collected and the centre boarded back up again.