"Entender la diligencia" commonly applies to service of notice. When notice is served, someone has to take delivery of it: not necessarily the person on whom notice is being served, but someone to sign or at least acknowledge service. "Entenderse la diligencia" can apply in that situation. Here's an example:

"El 23 de octubre de 2003, el Notario de Pozuelo de Alarcón, don Antonio Pérez Sanz autorizó acta de notificación de la escritura antes referida, en la que hace constar que en el día de la fecha, siendo las diecinueve horas y cuarenta minutos, se ha constituido en la calle P., n.º-, donde pese a sus repetidas llamadas no contestó persona algunas, ni se abrió la puerta y no encontrándose el conserje, vecino ni persona alguna con quien entender la diligencia."
https://www.boe.es/diario_boe/txt.php?id=BOE-A-2005-8336

This example is from Spain, not Mexico, but it's the same there. Here's an earlier question we had, in which the late Henry Hinds, who was very good at expressing Mexican legal matters clearly and accurately, explained that in "exhibiendo el original al X, con quien se entiende la diligencia, quien firma de recibido", "con quien se entiende la diligencia" meant "who is receiving notice or service".
https://www.proz.com/kudoz/spanish-to-english/medical-pharma...

In your case, however, I don't think we can use that translation, because it's not exactly a matter of service of notice. The "diligencia" here is an inspection. I think the word "diligencia" has the meaning "trámite de un asunto administrativo, y constancia escrita de haberlo efectuado" (DLE, definition 3).
http://dle.rae.es/?id=DmapFfE

So I think "entenderse la diligencia" means, in effect, agreeing that the proceeding was duly conducted. So I suggest "acknowledge the proceeding". In practice what must have happened was that an detailed official record (acta circunstanciada) of the inspection had to be made, and the "persona con la que se entendió la diligencia" was the person at the site who certified that the inspection took place.