When I called KBank customer support, the representative mentioned that I could submit it directly at the branch. However, when I followed up, the staff told me I should handle it myself via email.
Eventually, I didn’t submit the form at the branch in Phuket Town because I didn’t have my home country’s tax ID with me and I still wasn't sure if yes or no.
Providing my home country’s tax ID wouldn’t be correct, as I am not a resident there. I currently reside in Thailand (more then 180 days), but I don’t have taxable income here (yet.
Next week, I plan to apply for a Thai tax ID and eventually send the necessary documents. I suspect this will become increasingly important for dealing with banks, visa applications, and other formalities in the future.