Any research I try leads back to sites I don't know if I can trust, because most of them are random fact articles. I've found an official site that has some of the history, but not this specific fact. I also don't want to put much stock in wikipedia, because while it is useful it can be edited by anyone for whatever reason.
Well, as a hint, historically, the first people who'd have any interest in, or practical need for, writing down Japanese in the Latin alphabet weren't Japanese people. See the Nanban trade article in Wikipedia for a good jumping-off point to read about European contact with Japan. I'd bet you dollars to doughnuts that the first Portuguese sailors who showed up in 1543 would have made at least a few notes about the local language -- and by definition this would have been "romaji", after a fashion.
In terms of who made the first systematic attempt at writing Japanese in the Latin alphabet, that may have been a Portuguese priest by the name of Francisco Xavier in 1549. At least, so says this Encyclopedia Nipponica article (in Japanese). Frankly, I'd trust that more than the English Wikipedia article. Wikipedia is great for many things, but due to its open, anyone-can-edit setup, you sometimes get Randy from Boise posting all kinds of nonsense. Sometimes cross-checking against the same article in another language can reveal silliness -- I see, for instance, that the Japanese Wikipedia article on ローマ字 makes no mention of "Yajiro" in 1548.
I might be missing something, and someone named "Yajiro" may well have been the developer of the first romanization system. But from what I can find right now in a quick survey of materials, I don't think the English Wikipedia article is correct.
Update: A bit more detail about the English Wikipedia article Romanization of Japanese.
I did some digging in the edit history.
- The mention of Yajiro appeared all the way back on Halloween 2004 in this edit by user Sekicho.
- The edit includes no references and is unsourced, making it unfortunately dubious and useless for any further research -- we don't get any indication of where Sekicho got that information. For all we know, Sekicho heard that from someone in a bar one night.
- Additionally, that user's own page indicates that he is a native speaker of English, and an upper-intermediate learner of Japanese. This suggests that he may lack the skills needed to read Japanese-language reference materials.
In seeking to critically evaluate the source of the Yajiro information in Wikipedia, and the likeliness of that information being correct, the signs are not terribly promising. Considering also that the Japanese Wikipedia article disagrees, as does the professionally produced Encyclopedia Nipponica article, I must judge the English Wikipedia to be bogus.
(Full disclosure: I'm an editor at Wikipedia and Wiktionary. I have not previously edited the Romanization of Japanese article, but I will shortly to request a citation for this mention of Yajiro.)
Addendum: Sekicho's edit makes some historical mistakes, suggesting that the use of
⟨ f ⟩ in Portuguese texts was merely a spelling convention. Academic research into the history of the Japanese language instead interprets this spelling as indicative of the actual pronunciation of the time. So modern Nihon was probably pronounced as Nifon in the early 1600s.
There's also no mention of the important difference in treatment of long-"o" vowels. All of those that developed from //au// were rendered in the Portuguese orthography (spelling) of the time as
⟨ ǒ ⟩ (with a caron) and all of those that developed from //ou// or //oo// were rendered as
⟨ ô ⟩ (with a circumflex). Modern scholars interpret this as indicating that the two vowel values were different, with
⟨ ǒ ⟩ probably pronounced as //ɔː// (like English awww) and
⟨ ô ⟩ probably pronounced as //oː// (a regular "o" sound). These two merged into plain //oː// some time over the last four hundred years.