I think this would be the best term. It it sometimes referred to as "vigilante justice", in the sense of people taking the law into their own hands. This is from a book called Youth in Postwar Guatemala: Education and Civic Identity in Transition, by Michelle J. Bellino:

"Víctor shares a story about an instance of justicia propia (vigilante justice) that took place the previous night. A drunk driver hit a family crossing the street, and quickly a crowd began to gather, throwing rocks at the car [...]"
The driver was nearly lynched. He escaped by paying the authorities.
https://books.google.es/books?id=SG8kDwAAQBAJ&pg=PA118

One might sympathise with the people who wanted "their own justice" here, but I don't think it's appropriate to use a negatively loaded expression that amounts to mob rule. This really refers to private rule-based systems of justice outside the public state justice system, particularly practised by indigenous communities. The following refers to the kankuamas of Colombia:

"De todo este proceso, dos aspectos resultaron los más provechosos para las mujeres kankuamas y su organización: el primero, su insistencia en examinar de fondo la justicia propia de su pueblo [...]"
https://books.google.es/books?id=jXRQDwAAQBAJ&pg=PT433

As I say, I think "private justice" is a suitably neutral term which marks the contrast with public justice:

"Worried about the legal consequences, the comité cívico (civic committee) of Sarhuino migrants in Lima sent a letter to the president of the administrative and vigilance council disagreeing with the communal meeting's decision to practice la justicia propia (private justice, also referred to as 'social justice')."
https://books.google.es/books?id=MbSsk6thfGAC&pg=PA118

"The normative system of the Wayuu is a very superior system compared to the ordinary system. It is based on reconciliation and dialogue and differs fundamentally from the Western system of justice administration that is based on the figure of the ‘judge’. [...] It doesn’t have any costs – not even for the country - because it is a very effective ‘private justice’"
https://www.berghof-foundation.org/fileadmin/redaktion/Publi...

"El conocido sistema normativo Wayuu sobre el cual se ha diseñado un Plan Especial de Salvaguardia, goza del reconocimiento por parte de la UNESCO que lo ha declarado como Patrimonio Cultural Inmaterial de la Humanidad. [...] ‘No cumplir con el esquema de compensaciones ante hechos de violencia ocasiona desbalance en nuestro pueblo porque se constituyen en un irrespeto a la identidad lo que deja heridas en la memoria colectiva y tensiones internas que pueden llevar a la guerra, la cual -insistimos-queremos evitar por medio de nuestra justicia propia.’”
https://siic.mininterior.gov.co/sites/default/files/pueblo_w... (pp. 38-39)

The Wayuu are in northern Colombia and Venezuela.

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Note added at 2 hrs (2018-07-31 17:34:52 GMT)
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It seems to me that "justicia propia" is being distinguished here from "justicia comunitaria".

Is it certain that "justicia propia" refers only to indigenous justice systems? It seems to me that in principle it could include any judicial or quasi-judicial procedure separate from state justice.

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Note added at 2 hrs (2018-07-31 17:36:10 GMT)
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I should have said at the beginning of my answer that "justicia propia" can refer to vigilante justice, though I don't think it does here.