This is an area where grammar rules are not going to answer your question and you will be able to find plenty of uses of this construction either to suggest responsibility or not.
Plenty of people say “he got his car stolen” to mean that someone maliciously robbed him, and plenty of people would say “he got his nose broken” without the intention of blaming the subject in any way for his now broken nose.
The key is to be aware of the ambiguities of this construction and supply (or look for) context.
“Got” vs. “Had”
Unfortunately, there isn’t a strict, universal distinction between “got” and “had” in this case. Even worse, there will be people on either side with incompatible justifications for which word implies what fact(s).
Either of these two by itself could indicate scheming or innocent victimhood on the part of the subject:
He had his car stolen.
He got his car stolen.
My personal opinion is that the latter of the above two carries more risk of attributing at least some blame to the subject (see the next section of this answer or see Josue’s answer), but only if more context is provided can we be certain. For example:
Sadly, he had his car stolen last week.
Sadly, he got his car stolen last week.
He had his car stolen for the insurance money.
He got his car stolen for the insurance money.
Even in cases where it’s clear from context that the subject has had something stolen from them (i.e. taken without their consent), saying that someone “got” their car stolen can imply some responsibility on the part of the victim. Saying it this way can carry the message that the victim should have been more careful, or is experiencing the consequence of a known risk.
Part of the issue you are having comes from the fact that this is a problematic concept. Even though it’s clear in hindsight what the victim could have done differently, we must be careful about attributing a share of the blame to this person who had no intention of getting their car stolen. I’m sure everyone has had this experience of wishing we could change something we did after it turned out poorly in the end, even if the negative outcome was simple misfortune or external malice, and it’s not helpful to feed into these anxieties in others (especially as a casual implication without constructive feedback).
My advice is to avoid the ambiguity altogether. For example:
Someone stole his car.