L.,
God bless you for reaching out to this family. I was there 10 years ago, my husband passed away suddenly.
Meals, yes, I agree with the others, every other day is great, and continuing, tapering off, for as long as you can. As I said before, take one or two of the kids, let them talk, if they want but don't pry.
Dad may also need some adult time to talk, I know its hard but maybe a man in the neighborhood could make himself available.
I agree also that maybe one or two people can take the lead and let everyone else know what the family needs.
As bad as it sounds the family may feel some relief, living with mental illness is a struggle in its own, they may feel guilty for feeling relief. Let them know its ok to feel whatever they feel.
I found the hardest part is after the funeral, everyone else goes back to their lives, if you can continue to be there for them for months or longer.
If there are daughters they have specials needs. Kids identify with their same sex parent. Spend extra time with the girls, model for them what a woman/mother who is not suffering from mental illness acts like.
Best of all talk with dad on a regular basis and really listen. He may say something that gives hints as to what they need.
Give the kids cooking lessons, make it fun, but simple meals, spaghetti, sloppy joes, etc. Let them know its ok to laugh and have fun.
Check monthly to see if anyone needs help with homework. An extra pair of hands doing anything lifts the burden off dad and the oldest child. My oldest carried the burden of the responsibility even though I tried to not let him.
The oldest child may be graduating soon, a friend took mine and helped with all of the college details, what a blessing. He took him to visit different colleges, kept up with deadlines, etc.
Whatever each neighbors strength is, let them use it here. If one loves to play tennis they can offer to take one or two kids and teach them.
Hope this helps. Feel free to contact me if you have further questions.