If you have a court order stating joint legal custody, and the mother is the CP and the Parent of Primary Residence, then the PPR generally has the ability to control the times of appointments so that they are convenient for her life, and if necessary for her. However, you can ask the CP if she will accommodate your requests. You can also contact the doctor's offices and see if they will, at least, inform you when appointments are made so you may attend.
You need to retain counsel and have a consultation. You do not state whether or not there are any existing Court Orders or if you just have an informal arrangement. With Joint Custody, you have an equal say in everything that pertains to your son. Good luck.
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It sounds as if you need to consult an attorney to file a modification of the custody order so she lets you have more input in your son's life. A court order could help deter your ex from controlling the entire situation.
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If you have joint custody, you are entitled to full participation in the medical and educational activities of your child. This includes involvement in doctors visits and determining the medical care to be provided (or not to be provided). The school should also be notified that you should receive all notices that a parent would receive. If any of this is met with resistance, then, yes , you will need to go to the Family Part of the County and file an application to memorialize these rights.