I know that the term is used on the letter being written, but I'm not sure if it is required to use it on the envelope. Thank you.
You cannot use さん on envelopes. Just as you have to use Mr./Ms./etc on English envelopes no matter how you normally call them, you have to use 様 on Japanese envelopes. 様 is a more polite name suffix than さん. 先生 is also okay if you usually call them sensei (a teacher, lawyer, etc.). If you send a letter or a card to a kindergartner, くん/ちゃん is also acceptable.
Inside the envelope you can call them however way you think is suitable.
様 is for a person. You have to use 御中 instead of 様 when you send a mail to an organization. See: How to address someone in an email who referred to themselves as part of a department
Depends on to whom you are sending the letter. In business occasions it's very common to use 様 instead of さん because it shows more respect.