문화어 is largely based on the early 20th-century central dialect of Korean.
The distinction between 차 and 챠 (처 and 쳐, 쵸 and 초, 자 and 쟈, etc) were already long gone in the central dialect (which includes Seoul and Pyongyang) in the 20th century.
Before Korea's division, in the 1933 orthography reform (한글 마춤법 통일안), which both 표준어 and 문화어 standard orthographies are ultimately based on, abolished spellings such as 쟈 and 챠, and replaced them with 자 and 차, because they were no longer distinguished in the central dialect.
So it makes no sense for 문화어 to have '자동챠' instead of '자동차'. In fact, it makes no sense for native and Sino-Korean words in 문화어 and 표준어 to contain syllables like 챠, because they were all merged with 차 in the 17th-18th Century. (Note that in dialects further up North, specifically in the Northwest, 자 and 쟈 etc were not merged and are preserved. However, this has little to do with 문화어.)
It is often said that if a South Korean says "자동차", a North Korean will hear "쟈동챠".
Then why does this happen? This is because in North Korea, 'ㅈ,ㅊ,ㅉ' is more often pronounced de-palatalized before back vowels (ㅏ, ㅓ, ㅡ, ㅜ, etc). This is a relatively recent phenomenon that happened in the late 20th century.
Note that this phenomenon is not NK-only, it is also happening in the younger generation in Seoul as well, especially among female speakers.
This means that the phonetic distinction between 차 and 챠 are coming back. However, distinctions that were already lost in the 17th century cannot revert themselves, so 차(車) stays as 차, instead of reverting back to 챠 as in Middle Korean.
In fact, synchronic contractions such as 치여 -> 쳐 "hit" are contrasted with 처 "wife" in 문화어, as well as in Seoul's younger generation speech. However, this distinction is not yet accepted by South Korea's 표준어.