It sounds like you have to continue to parallel parent.
If the children are old enough, tell them to call you. If you have any sort of system where you can view your child's records from the doctor (our HMO has this) make sure you are on that so that you can look up when your child was taken to the doctor. Follow up with the doctor about the diagnosis. In our case, we can email the doctor's office for additional info.
If the coach is not yet aware, talk to him /her /them about your child's participation. There may be a time where the child himself needs to choose to participate or not and potentially ask his father to miss time...BM gave my SS that ultimatum and then was angry when he chose sports. So she didn't see him at all for the whole season.
You don't need to get along well with him. You just need to be civil. Give the kids OTHER role models for a healthy relationship - friends, grandparents, aunts and uncles.
Try not to badmouth their father, but also try not to completely cover for him. "I don't know" is a valid response. If you have court order about communications, then bring that up to your ex. If their father is abusive, then get them into counseling (might not be a bad idea anyway), where they can deal with their issues in a safe place.
You might also consider carpooling or hiring someone responsible to take your son to sports practice in your town. Sometimes to get things done we just didn't factor their mother at all. Will she help? Highly unlikely. So we just go do it ourselves. She always signed the form, but we always knew we were doing the schlepping.
You should also document everything (without malice) just "called the children on x day and time. xH did not answer the phone or allow me to speak to them. Sent email on x topic (attached) on x day and time. Per readnotify, the email was read on x day and time. As of x day and time, there has been no response."