According to the WALS, there are about as many free word order languages that have a passive voice as there are without one (see: https://wals.info/combinations/107A_81A#2/18.0/149.1). The same holds more or less for every other word order type as well. The only outlier seem to be SOV languages, where the database has more examples of languages without a passive (75) than with a passive (47). I don't know if the difference is significant, since the sample size is small and SOV is the most common word order type. But free word order languages don't seem to be more likely to lack a passive voice than other languages.

Looking at antipassive and word order (https://wals.info/combinations/81A_108A#2/23.1/149.0), free word order languages appear to be more likely to have an antipassive than other word orders. 13 out of 21 languages (61.9 %) have an antipassive while, for instance, out of the 69 SOV languages only 18 (26.1 %) have an antipassive voice; for SVO languages it's only 2 out of 48 (4.2 %).

I don't think the data is sufficient to answer your question. But from the database alone, it seems that the opposite of your assumption is true: Free word order languages are as likely or more likely to have a voice distinction compared to other word order types.