I once bought (but in the end chickened out of sending) a card that said on the front "I'd like to be the first to say I'm sorry" - those last two words were really large, so at first glance the front just says I'M SORRY. Inside it says "And I'd like you to be the second." And there is plenty of room for a note.
While I didn't send the card (demanding an apology felt wrong to me) I think it's a reasonable approach. Something has happened between you and this other person, and you can't let go of it, you're the "wronged person" and you really want to hear that apology. So take a good hard look back at the interaction. Did you get super angry right when the person transgressed and react loudly or rudely? Have you been avoiding the person ever since the incident? Note: I am not asking you to consider apologizing for your role in the incident. I presume that if you felt the incident was partly your fault, you would have apologized for that already. I am talking about afterwards, during this period of, well, sulking, or at least reconsidering the relationship.
You can apologize to the transgressor for how you reacted to the incident. Something like:
I am sorry for yelling at you on Tuesday. When you [whatever] it took me back to some dark times in my past and I was yelling at my old high school bullies [or my ex wife, or my crazy former boss, or my father] more than I was at you. Saying what I had always wanted to say when they did [whatever.]
What you did was not ok. My reaction to it was also not ok, and I'm here to apologize for it.
Most people will almost instinctively apologize when they are apologized to:
Oh my, but no, I should apologize to you, I should not have [whatever] and I realized right away that it was wrong...
But even if they just smugly accept your apology and don't apologize themselves, you have started the conversation. You can now carry on to elaborate on why [whatever] wasn't ok. You can also say things like:
It would really help me to know that you agree that [whatever] was the wrong thing to do/say and that you regret it, or will try not to repeat it.
Asking people specifically to apologize could take them back to childhood incidents where they were forced to apologize when they had done nothing wrong, such as to abusive teachers or a bullying older sibling who teased them into hitting. Asking them to say they understand they caused you pain and will try not to do it again is far more useful for you emotionally, even if they never actually say "sorry".