What is the first thing a missionary has to do when they go to a new land? Translate the bible in the local language! Literacy is closely tied to proselytization. Portuguese, Spanish, and French missionaries arrived in this area of mainland SE Asia in the 16th century. At that time, and until the early 20th century, only the mandarinate elite of Vietnamese were literate. Why? Because Vietnamese was written in an alphabet (ED: as /u/keyilan notes, it's a writing system not an alphabet) called Hán Nôm, a complicated system that used Chinese characters for Sino-Vietnamese words (many words are rooted in Chinese) and adapted Chinese characters so that they could represent Vietnamese words. To learn this system required years of study with a Confucian scholar-tutor, who himself had spent years studying, only to fail the imperial mandarin exams. Hence why he was teaching children, not head of a district.

To overcome this literacy gap, missionaries devised a new Roman alphabet that represented the Vietnamese language in Roman characters, with the addition of diacritic marks to represent the tonality of Vietnamese. This alphabet is now called quốc ngữ (‘national script’).

The promotion of the new script came from French colonial and Vietnamese sources in the early 20th century. A colonial government needed plenty of translators and functionnaires/bureaucrats to operate. They needed to know French. Instituting quốc ngữ allowed students to learn Vietnamese much faster and due to its similarity with French, allowed them to learn to read and write French quicker than before. On the Vietnamese side of things, people alike saw that quốc ngữ could expand literacy. Reformers simply hoping for the betterment of society saw literacy as a means to spread new ideas to the public. And revolutionaries wanted more people to learn to read and consume radical theory. Even the famous Phan Bội Châu, a Confucian trained revolutionary who could only write Chinese and Hán Nôm, encouraged younger generations to learn the roman script. By the 1920s and 1930s, there was a thriving Vietnamese print culture in large cities across Indochina. After 1945, both the communist led Democratic Republic of Vietnam government and French sponsored State of Vietnam, launched literacy campaigns to teach youths and adults alike how to use quốc ngữ.

One last thing, this process was not unique to Vietnamese. Even in the mid 20th century, missionaries were helping create new Roman scripts. The widely used Hmong RPA script was the work of French and American missionaries and Hmong leaders.

The short answer, at least for how they got the alphabet: colonization. The writing system for Vietnamese was switched to Roman characters largely because of the French colonial administration, from a system devised by earlier missionaries.

Now, as for why they adopted it wholesale when others did not, perhaps someone with a better hold on Vietnamese and Asian history can answer that. I suspect it had to do with ruling elites' willingness to keep it. Turkey is another example of replacing an alphabet in a region where most did not: Ataturk imposed the Roman alphabet (replacing Arabic script) from the top down as a way toward his vision of progress and Western secularism starting in the 1920s.

Like the rest of the cultures that fell under China's cultural influence, Vietnamese actually did use characters at an earlier date before switching to the Latin alphabet. The system was called Chữ nôm ("common characters") or Quốc âm ("national speech").

Some of the characters were taken directly from Chinese, but some were local inventions such as the number three, which is sam in the Chinese of the time but ba in Vietnamese. The sam character is 三, and 巴 was pronounced ba so the Vietnamese character was 𠀧, pronounced ba and meaning "three". Many of the native characters are of this sort, taking the approach of a mix between semantics from Chinese and phonological representation of Vietnamese. Korean actually did the same thing with their own locally invented Chinese characters, so this isn't in and of itself all that surprising.

Despite being in use for about almost two millennia, it wasn't until the mid 1800s that these characters were truly standardised, and then in 1910 they officially officially switched to the alphabet they have now (though it took time to take hold), which as /u/bclelandgt mentioned was the result of Missionary influence. The new alphabet was called chữ Quốc ngữ, also effectively translatable as "national language script" (written 𡦂國語 in the Chinese character system), often abbreviated to just Quốc ngữ. In 1918 the emperor officially declare the character-based system done with, and a decade or so later the characters had finally truly died out.

Wikipedia actually has some pretty good articles on the relevant topics (which is always nice). You can read about the writing system of Vietnamese more generally here, and about Chữ_nôm here.

Let me know if you have any follow ups.