I didn’t verify this yet, but I heard that part of asking for mechila is the embarassment to have to ask them for mechila. Therefore it’s better not to ask for mechila in a letter/mass text/email. I am a fan of a mass text just saying Gmar Chasima Tova, etc. It shows that you care about them.
I actually heard a nice vort on this topic. During the Asseres Yemei Teshuva, people try to do more teshuva. They keep pas yisroel, chalav yisroel, ask pople for mechila. What happened the rest of the year? The whole year you need to do teshuva! The answer is that imagine that we wouldn’t have the asseres yemei teshuva. There would be no aliyah whatsoever. Therefore we have an asseres yemei teshuva and that should take us through the whole year and remind us to do teshuva if we do an aveira