I don't think vowel differences are sufficient to deem a language impossible to romanize. Japanese has been romanized effectively for decades despite the fact that Japanese vowels are quite different from English ones.
I think this comes down to two competing approaches in transliteration:
1. Transcribe the original non-roman text into something that a native English speaker would read correctly without help. This never works well due to vowel differences and accents, but nevertheless it works well enough that it's very popular. An example of this failing horribly:
Hiroshima: often pronounced Hih-RO-shih-mah. Actually pronounced Hee-roh-shee-mah (nevermind the Japanese R).
2. Romanization without intent to be pronounceable. Nihon-Shiki romanization seems to use this approach: it's unreadable by English speakers, but unlike other romanization schemes, provides a one-to-one mapping to the original kana, i.e. it loses no information.
I think it's important to note as well that "roman letters" != "English"; Nihon-Shiki is an example of this, the creation of roman letters intended to be pronounced by non-English speakers.