I totally get it. I would not like anyone dropping in unannounced especially on a day with one sick kid and somewhere to be by a certain time. However, if your husband can cool the seething for a while, maybe you both can think through how another generation sees these things.
I'd ask your husband --when he was growing up, were there relatives or close family friends who just dropped in on his parents like they drop in on him? When his parents were kids, did folks just drop in on each other (I'd bet they did, back then)? That could explain why they think it's OK -- if it was normal for them. I am not saying it's an excuse; of course they should call and not ever drop by unannounced; but it may help you both understand why they do something that is rude to you but apparently not to them.
Also, her four calls in one hour probably mean she was thinking, "I DID call in advance" and she probably was upset that no one picked up, especially when she got there and found you all going about your business; she likely thought, "They were right here and never picked up and I called and called; they were avoiding me, etc." Yes, she did not get the "If we don't answer it's a bad time" rule, but to her, she may have been worried that no one picked up -- "Is something wrong over there? I thought they were home..." This is how many of my late mom's generation would see it. Again, not an excuse, but depending on her age and background, she may feel that the phone is something one ALWAYS answers "in case something's wrong." Some people of our parents' generation, depending on their backgrounds and how they were raised, often do think every call must be answered and they answer every call they get, so they don't really understand why someone would not pick up the phone if they're home.
Also, is she just lonely and bored? You said "they" so I assume his FIL is around but she may be lacking anything of her own to do, if she drops in on you when she likes. Steer her toward some kind of activities or make a proactive effort to take her with you somewhere with the kids along; she may be feeliing that as the kids have gotten older, she sees less and less of them, hence her wandering over. Again, not an excuse, but if she's not getting the "please do not drop in" message, she may be using your family to fill a void, and/or she may be wanting contact with your family because she knows the grandkids won't be there much longer.
I don't know her so I can't say if she's toxic, obnoxious in other ways, etc. But if she's otherwise OK, other than the dropping in, why not work gently with her on it? Why not tell her, "We know you got upset and we don't want to upset you. We do want to see you! But it was a really tough day (sick kid, place to be, etc.)" Then maybe set up a regular time you get together with her and FIL, a standing appointment to DO some activity everyone can like together (do they bowl, like to eat out, go for walks? Do they want to go to the kids' school or sports activities, and do you ever invite them specifically to come to those? etc.). If you already do all this and see them like this, well, this is all moot. But if not, then more invitations from your family to do specific things together, coupled with firm but kind "not dropping by" policy could help alleviate things.
And if your husband's, your own and your kids' relationships with your MIL and FIL are really otherwise OK, work on this ,but please don't let it force you cross-country. Having in-laws who are nearby could be a good thing in many circumstances if you need them. I wish my own mom could drop in (though yes, she would call!).