I have a different view, and this is a very similar situation to my own. I hadn't seen my son for 5 years and have started to again just a few months ago.
The ex has told him to write, she hasn't stipulated that this needs to be a letter.
At 13 your grandson will have a phone, and probably snapchat and Instagram. I joined Instagram and made contact through there with my son.
For me it was odd, I wanted to contact my son as my wife is pregnant and I wanted him to hear from me, but I had no way of making contact, he popped up on my instagram as someone I may know, so I sent a message and got a reply.
Your son may not get a reply or he may not get one that he wants to hear, but at least your grandson will know your son wants to be in his life and cares.
Advice I would give though is if contact is made, and conversations start, the subject should be hadnled carefully, no matter what the ex has done, your grandcild is caught in the middle and doesn't need to hear that all the blame lies on his mum (even if it does)
We don't have a number for my grandsons mobile although I am sure he has one. We looked on all the usual social media (snapchat, facebook, instagram, twitter) but if he is on there it is not with a name we can work out is his.
We have been given an email address for him but it doesn't link to any social media so we guess it has been made specially for us.
I am worried about my sons mental health, but I think if he doesnt try that will also be bad for him. Lose lose situation.
I was hoping that if we supported him through the court route, we would be able to get somewhere and perhaps although he would need to write first that we would be able to get support in making him read the letters. It doesnt look like that is the case so we will jump through whatever hoops are put in front of him.
The fact that his ex is even pressing for him to write is progress and a slight mellowing from her as previously she wanted him to have nothing to do with him at all. She has also been willing to speak to me (but not my son) and we have had a meeting)
We are taking out a magazine subscription for my grandson for one of his hobbies (suggestion of his mother) and she has also suggested that when the subscription for his hobby fees are due next year he could pay for it and she will tell son dad pays for it.
Perhaps she genuinely has mellowed and writing will lead somewhere. I will keep you all posted.
That's a shame that you can't find him on social media, as that would have been a good route to go directly to him.
Maybe the mother has mellowed, I know my ex seems to have and even drops my son at my house, allows him to stay as long as he wants, and even come short notice. all things that would never happened 5 years ago.
Hopefully your sons ex has as well.
It sounds as though you are doing all you can and you are supporting him, I know it's important to feel supported and how much it means to know someone is in your corner.
Could your son suggest that he supplies his son with a cheap mobile phone so that they can keep in contact at pre-defined times? The mother has the option to switch it off between times (and I would refrain from texting anyway without her permission to build up trust) - just a thought.