My suggestion:
Through the composing of a proceedings / Through the assembling of a minutes (record). Regards and thanks.

I don't think that with the use of the word levantamiento the author meant anything other than "writing" (or "drafting"). However I agree with Answerer Bowley that "filing" (or "submitting") is understood. Whether the report is deemed "formal" depends on organizational rules etc. which are not disclosed as context by the Asker.

Note that recordation in US Am. need not refer to an escrow 'deeding' and registration of land - see the Merriam Webster ref., plus minuting cf. Ninoslav might be deemed incongruous outside of a construction reporting and company meeting context.

A legalistic turn of phrase like 'engrossment of the record' is bound to attract howls and squeals of protest against archaic legal English, plus an 'engrossed Bill' in the US Senate - as is well-known in the UK - sends the meaning a different way.

"levantamiento del acta" just means writing/drawing up a report. This is just a very long-winded way of saying that any loss or theft of assets must be (officially) reported to the relevant authorities (i.e. the police or in Mexico at least, el Ministerio Público).

This whole section
"En los casos de robo o extravío de un bien, el usuario deberá denunciar el caso ante las autoridades competentes a través del levantamiento del acta..."

can just be translated as:
Users must report any theft or loss of assets to the competent authorities

...and notify the manager [or whatever you're using for "administrador"] in writing so the necessary steps may be taken.

For "autoridades competentes" I usually use "competent", but "relevant" and "responsible" could also work.