Your sister should check with her divorce attorney because the problem revolves around custodial vs non custodial parent. "Residential" parent really is not an issue here. Remember that secretaries are essential to a school but they too make mistakes. Just to make sure your sister should get the final(and accurate) word from her attorney. It should just take a quick phone call to clear up the "language" of the situation.

Also, the school needs this information. If they don't have the correct information or just don't have the information about custody at all, they will be very cautious. A school needs to be very careful about protecting their students. If your sister indeed has custody and the school has this information then she should absolutely be able to talk to the teacher.

I'm a bit confused about the fact that the children are with their dad the entire school year, but you state your sister has equal custody. I know there are many circumstances that can be the reason for this, so as long as she truly has any amount of custody, the teacher is wrong re: getting all info from the father, and this is ABSOLUTELY ILLEGAL! As another writer stated, yes, she must by law provide documentation to the school supporting this. That's unfortunate as well, because divorces also leave you vulnerable to having to divulge a lot of things that quite frankly are not the business of the school and faculty. That is the law however, so we have to comply.
The school SURE does want to protect themselves, but that's what they have attorneys for. Since sadly, but realistically this is a common situation, they certainly should know the laws associated with this.
What i KNOW factually is, the ONLY situation in which a school can refuse to allow a parent to pick up their child, or refuse any/all info, is if there is just cause. Such as, the parent is not permitted by court order to see the child, the parent only has supervised visitation, etc. AND the other parent provides this to the school. This is not the case here, so shouldn't be an issue.
I'm giving you some info that probably doesn't apply, but i wanted you to know the law in more depth. In the event your sister cannot or chooses not to use an attorney, she can provide this piece of info verbally at least.
Lastly, you should know, that not only is she permitted to know every bit of info that her ex-spouse receives re progress, the school is obligated by law to see to it that she also receives EVERY piece of information re: school events, happenings, when conferences are being scheduled, etc.  If it comes to the home via email, she needs to request her email address be added as well. If she does not have inter-net access, they must send paper copies to her, or contact her via phone. She can legally request a copy of her child's report card also be mailed to her, if her ex currently receives it via a backpack. The school must do this. As long as you are pleasant with requests, one should never be afraid to ask for ANYTHING they want or need to know. If they don't comply, you must then demand it. I have many other examples, but to sum up, whatever pertains to her situation, & she does not have access to via her ex, she has a right to and whatever way she needs to receive it, is what the school must supply.
She is being mistreated by either uninformed school staff, or someone there who doesn't want to do some things twice.
I know all of these things because my ex spouse receives mailings, e-mails, all info i do. The school also has a website here to access a lot of school info. I'm totally fine with the laws, as your ex may be & i never had a reason to intervene. In my situation, we are in separate states, so the mailings & such are sometimes necessary. Other times, i provide what i think he may not have received, during visitation times, or ask if he's aware of the major info at least.
I hope i have helped. There is a lot to know & if you aren't educated, life can be grueling & stressful.
All of the best to you to correct this problem.

Some of the answers were very long so I did not read all of the details. To make it short, your husband or you must submit the divorce papers to the school showing that you both may have access to the records. The school can not (and the parent's should not want) let just anyone have access to their child(ren). As it is, only the parent who enrolls the child will have access and anyone they put on the enrollment paper work. If your ex has not done this, then it is up to you to submit the proof that you share custody and are allowed by law to receive the updates and progress of your child. There are many parents out there who do not have custody and by law/protective orders are not allowed to vist/know about their child(ren). First and foremost, the schools responsibility is to the child's education and safety. I personally would be very upset to find out my school gave out information about my child without my permission just because someone says they are the shared custody parent. Anyone can say it. You have to prove it.

Just to clairfy, I do know we are talking about your sister. I was just trying to put an answer in first person.