Sending and Receiving Email in Japanese
Most email programs and webmail programs in use today should support Japanese. Here I'll describe how to do this with Apple's Mail application.
Japanese Email with Apple Mail
To support Japanese, an email program has to be able to enter Japanese text as you compose the message, then convert the mail to a common Japanese encoding when it is sent, and finally read mail in all the commonly encountered Japanese encodings. Whether your mail is readable by the recipient depends to some extent on compatibility between what your program sends and what the recipient's can read. In most cases, mail should be able to do all this automatically without help.
Mail used to have a menu that allowed users to troubleshoot problems by manually resetting the encoding for outgoing mail messages or changing the display encoding for incoming ones, but that menu has been removed in more recent versions of MacOS. So if you do encounter cases where incoming Japanese mail does not display correctly, or the Japanese mail you are sending is unreadable to your Japanese recipients, there does not seem to be a simple straightforward fix within Mail. You might try webmail as an alternative in those cases.
Webmail services like Yahoo Mail and Gmail not only support Japanese, but also have Japanese interfaces available if you want to try them. See the Web Browsing section of this site for tips on displaying Japanese in your browser.