Your company typically provides the business cards for you. Even in eikaiwa or JET, your company or board of education should supply your business cards, as you are representing their organization. The purpose of the business card is to identify you with your employer, not to announce you as an individual. The exception is if you are an individual entrepreneur, or an artist.
If, for some reason, your employer does not make business cards for you, you can make one with just your name and your occupation on it (and not make any association with the company or organization). For example, if you are employed by Interac (used to be a well-known eikaiwa agency...I don't know if they are still around) and they do not provide you with cards, you can make simple cards with your name and private contact details, with the occupation simply as "English Teacher" or "assistant English teacher" on it. No need to put anything else on them - just your name and contact details (yes, email included, and website if you have one and it is appropriate to your occupation). Qualifications are also OK (Phd and whatnot).
As you guessed, business cards are super-irrelevant in this age of remote learning, remote-working, remote-everything. But, yes, people still have them.