A.K.
I wouldn't worry too much about it. You already know she's touchy and it sounds like she's raising her children that way as well. I have someone similar to that that I deal with and I just flat out tell her that certain things are just without negotiation. Don't get into arguments about parenting because noone ever wins those. :(
For your SIL as well, just steer clear as much as you can until she has calmed down. Then try getting her one on one to find out if there's something going on that is getting her so upset. You might even try opening with concern over her child. Maybe that child has been sick or recently been picked on and she's just overly concerned? See if you can get back to agree-to-disagree and just stay calm for everyone's sake.
You have EVERY right to insist on safety! What would have happened if the police pulled you over and saw that you had an improperly restrained kid? Or, God forbid, you had an accident and that child was severely injured or killed while the rest walked away? Based on what you have said, I think you did the right thing.
For the future, even though you have a van it certainly isn't a license (or requirement) to become the kiddie chauffeur. I'm sure you didn't mind the extra company, but it may be time for a polite "I'm so happy you would like to ride with us, but we have some important van rules to keep everyone safe. If you aren't sure you can follow these rules, please ride with your mom/your dad/your sister/someone else." Ahead of time, work with your kids to figure out the major rules you have and STICK TO THEM.
Good luck!