Both four-door car and four-doored car are correctly constructed according to English grammar rules. Which one is more idiomatic based on actual usage is a separate issue.
The use of a phrase four-doored car does not necessarily imply the existence of a verb to door. English -ed has multiple functions. It can form the simple past tense of a verb, as in They proved it. It can form the past participle of a verb, as in I have watched it before. Some adjectives, such as excited, are derived from a verbal past participle ending in -ed. But a third function of -ed is forming adjectives directly from nouns (or nominal phrases) with the meaning "possessing [noun]".
Some examples: small-minded, two-faced, cross-eyed, brown-haired.
These are all idiomatic and usual, even though it is not usual to say *"They are minded small", *"They are faced two", "They are eyed cross", *"They are haired brown". Also, there is no implication that the state is a result of some change, as there often would be with a participial adjective (describing someone as brown-haired does not imply that their hair became brown after previously being a different color).
The Oxford English Dictionary has a separate entry for this use of -ed, which says the precise details of the etymology are unclear. Some quotations:
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In modern English, and even in Middle English, the form affords no means of distinguishing between the genuine examples of this suffix and those participial adjectives in -ed [...] which are ultimately [from] nouns through unrecorded verbs
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The suffix is now added without restriction to any noun from which it is desired to form an adjective with the sense ‘possessing, provided with, characterized by’ (something); e.g. in toothed, booted, wooded, moneyed, cultured, diseased, jaundiced, etc., and in parasynthetic derivatives, as dark-eyed, seven-hilled, leather-aproned, etc.
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Groundless objections have been made to the use of such words by writers unfamiliar with the history of the language
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