The name order isn't your only issue to worry about. A Japanese system might be expected to have two fields, one for the name as it's written and another for pronounciation. Just in case someone decides that 小鳥遊 as a name is pronounced as たかなし.
I remember Amazon.co.jp having two name fields when you registered a new user, and in English they were called "Your name" and "Your name again", because a name pronounciation field probably makes little sense as a concept to most non-Japanese speakers.
Well there are some kanji names that have multiple pronunciations (with only one being correct for an individual). And some Chinese people in Japan go by the Japanese pronunciation of their hanzi/kanji name.
They way these fields are done in Japan is often difficult if you have a non-Japanese name and give a lot of insight to this discussion. I've encountered many variations of these two sets of fields. Usually the name fields accept kanji, hiragana and katakana, sometimes accept roman letters, but sometimes if the site can be set to english, only then name fields may or must be entered in roman letters. Usually the pronunciation fields must be in katakana, but sometimes they must be hiragana and again sometimes if the site can be set to english, it may or must be entered in roman letters.
Sometimes, like for a credit card application, there is a third set of fields for romanization. And for names, romanization has standard rules but there are exceptions such as when your family has been using a previous romanization so you shouldn't automate it, especially if an exact match to a passport or credit card name is required. Many other languages have different romanization possibilities for names too. There are even cases where bank insists on one romanization and the passport office insists on another leading to airplane tickets issued with the wrong one!
Some (mostly old) systems require so-called full-width roman characters (same width as a Japanese character and monospace) and some systems require so-called half-width katakana. The latter usually doesn't accept small katakana or diacritical marks, making it difficult to read and write non-Japanese names (or company names for that matter). Back in the day to transfer money at a bank you had to successfully write the recipient's name this way but these days all banks can automatically grab the katakana name and you just have to check it (but it's still in hard to read half-width katakana).
A common annoyance is when everything must be in full-width characters except phone number which must not be.
Finally, there is one type of field that really can trip you up: the address field. It usually must be in full-width characters which is fine in principle, but building names are often (officially, on the record) in English. The windows Japanese IME will happily default to "normal" and not full-width characters. This is easy to spot for a building name and figure out your mistake. But if you live on the third floor (denoted by adding "－３Ｆ") it's hard to spot an accidental "-3F", "－3Ｆ", "－３F" or especially "-３Ｆ" (the dash is wrong).
I remember Amazon.co.jp having two name fields when you registered a new user, and in English they were called "Your name" and "Your name again", because a name pronounciation field probably makes little sense as a concept to most non-Japanese speakers.