Briefly, since my DBIL and his partner split, partner severely restricting access for my neice for our side of the family. I have read about applying to the family court for a contact order for my DFIL, but we need to prove he is of a benefit to my neice's life. I have questions. 1) what evidence is required? 2) Do aunts and uncles of the child have to apply to the court for a separate order? If not, and we want to see the child against the mothers wishes, what options are there? 3) how much may the legal costs be? We don't have a lot of money in the family.
It's not up to her who BIL takes his daughter to see on his contact time. He should facilitate contact between his daughter and his family then if BIL's ex has a problem with that then they should go to court for a formal agreement.
Usually Court applications for contact by a grandparent are made when both parents are against them having contact. IF they are awarded contact it is likely to be minimal, a few hours a month for example. You can't then have various aunts and uncles all making applications for contact time too, it's just not done that way.
If DBIL has an informal contact arrangement with his ex then he should use that time for his daughter to have a relationship with his family. If his ex is demanding he doesn't do that then he needs to attempt mediation and then apply to Court for a proper defined contact order which means she cannot control his time with his daughter.
This will be questioned by the Judge and in my opinion FIL making an application for contact is going about it entirely the wrong way given the child's Father, his son, who has contact with the child isn't against him having a relationship with the child.
In any case, FIL needs to attempt mediation before applying to Court as otherwise his application will not be processed.
I suggest seeing if you can mend your relationships with the mother directly. Much cheaper than court and much more flexible and better for the child than having their mother spend time and money fighting a court case.
Has the mother said why she doesn't want contact with the extended family?
As has been said up thread, she cannot control who the children meet whilst they are with your DBIL. If she insists on putting conditions like this on the time they spend with him he should apply for a contact order. That will be quicker and easier than your DFIL applying for contact.