You got the idea. Actually it mean both: automatically (by the rules) and straight away / immediatly. There is nothing to say, and no matter what happens around it, you won’t be able to get back in the competition.
It emphasises the violent aspect of the disqualification (without negative connotation). To give a precise example about Judo, rules recently changed so one must not put his hands on his oppenent’s pants. If so, he is “disqualifié d’office”.

I think you may use “direct” or “directement” as équivalents as follows (from less formal to more formal):

Si tu fais mal à l’adversaire, tu es disqualifié direct.
Si tu fais mal à l’adversaire, tu es disqualifié d’office.
Si vous faites mal à l’adversaire, vous êtes directement disqualifiés.

An other context where you use “d’office” would be “un avocat commis d’office”, where the law states that if you don’t want to pay for a lawyer, the State will assign a free one to you. I don’t see any other usage of “d’office”…

d’office:

• automatiquement
• en vertu d’autorité
• en vertu des obligations

A l’origine, il s’agissait d’une charge, d’où les sens précédents.

voir le CNRTL