Learn Spanish (Latin America): ¿Tramitan la hipoteca con el aval con quien hablé? - Do they process the mortgage with the guarantor with whom I spoke?

It is an implied "they" (ellos or ellas). In contexts like banking or administration, Spanish often uses this plural form to refer generally to the institution, the bank, or the staff processing the paperwork, much like saying "Are they processing the mortgage?" in English.

What exactly does aval mean here?

In banking and real estate, an aval is a guarantor or co-signer—someone who legally agrees to take on your debt if you can't pay. While aval can sometimes refer to the financial guarantee itself, we know it refers to a person here because the sentence uses quien (who).

Why are there two con's so close together (...con el aval con quien...)?

Each con belongs to a different part of the sentence. The first one means "with the guarantor" (processing it with them). The second one goes with the verb hablar, because in Spanish you speak with someone (hablar con alguien). In English, we do the exact same thing here: "...with the guarantor with whom I spoke."

Could I say con el que instead of con quien?