Learn Spanish (Spain): Denegaron la instancia, puesto que no adjunté el DNI. - They denied the formal request, given that I didn't attach the ID.

Formal causal connectors like puesto que, ya que, and en vista de que are used to state the objective reason or cause for an action. Because they declare facts, they are followed by the indicative mood. Here, no adjunté is the factual reason the application was denied.

Why use denegaron here instead of a more common word like rechazaron?

Denegar specifically means to formally deny a request, grant, or permission. It is highly typical in legal and administrative contexts. Rechazar is a more general verb meaning to reject, such as rejecting an offer, a gift, or an idea.

What exactly is an instancia? Could I use petición instead?

You could use petición to generally say request, but instancia is the precise administrative term used in Spain for a formal, standardized written application submitted to a government body or public authority.

What does DNI stand for?

It stands for Documento Nacional de Identidad. It is the standard government-issued ID card in Spain. Notice that in Spanish, acronyms like this are almost always preceded by the definite article, as seen here with el DNI.