As long as you have the cash you should in theory be able to walk in to most Japanese banks and ask to send a 振り込み (furikomi) as long as you are carrying hard currency. (If you are sending to a post office bank account better to walk in to a Japanese post office during business hours (郵貯銀行）.

Banks business hours are 9am and to ensure the furikomi is sent that day most banks have a cut off of 3pm.

However be aware you will be charged anywhere between 300 yen and 1000 yen depending on if
a) the suppliers bank account is the same bank
b) the amount.

My best advice would be to find a bank employee and say

銀行に振り込みたいです。すみません日本語が話せないので助けてください。
ginkou ni furikomitai desu. sumimasen nihongo ga hanasenai no de tasukete kudasai.
I want to send a money transfer. I can't speak Japanese so please help me.

And then show them the piece of paper with your suppliers bank account details (and of course the cash) - (written in Japanese)
You will need to send it
- your details
- your suppliers bank account details
- your suppliers address
- your suppliers contact details

-mrploddy