History teaches us that any language can be written with any script.
Missionaries in China have written various Chinese dialects in Latin script. The most successful one, still used today by Christian missionaries, is the Peh oe ji (Minnan). Then there's also Hakka and Mindong on Wikipedia. There's also Shanghainese, Suzhounese and various Zhejiang dialects written in Latin script.
Korean has a phonemic script, although it has a lot of homophones.
Vietnamese has even more tones and more vowels than Mandarin, yet it's written in Latin script, and there are still homophones.
And sure, Shanghainese has also that problem of not having characters for some words.
Most Chinese dialects don't have a literary tradition, with few exceptions. Like 《海上花列传》, a novel written at the end of Qing dynasty, written in old Shanghainese.
Nowadays there are many linguists trying to finding the etymologies of the words and using characters from Classical Chinese (the so called 正字 - proper characters) or creating new ones (Cantonese created many of them).
But even in a Shanghainese forum, where they try to use 正字 as much as possible, they have many doubts, or sometimes they transcribe that word in romanization.