It doesn't matter what she believes. She isn't the parent. Her belief system, therefore, is not relevant to your child and their welfare.
Although sun-burn doesn't necessarily lead to skin-cancer, it is one of the factors towards its cause in (too) many cases. My father, who grew up in a time before sunscreen was even thought of, has two worrisome blemishes which have suddenly appeared on his face. He's terrified that, because he spent his entire early life outside in the sun, "running around without any protection from the sun", that he has skin-cancer. It is a possibility. He's off to the GP this week to get them looked at.
In comparison, my son is actually allergic to the base ingredients in every sunscreen/cream/lotion we've ever tried on him (he must have been one of the only babies who was allergic to Johnson's hypo-allergenic baby products, for example!). Luckily, he has olive skin and doesn't burn - but when he was younger (he's 16 now), it wasn't through lack of his trying, I can tell you! Both my ex and I made sure he had hats with flaps at the back, to protect the neck (or he wore them backwards so that the solid front of a baseball cap covered the neck), long loose-sleeved shirts on, and he rested in the shade during the hottest parts of the day. Ex-MIL kept trying to slather him in lotions (which came from a good place - she wanted to protect him as she thought we weren't already following medical advice on his being in the sun) and couldn't understand why he'd start to scratch within seconds, come up in hives, and run a temperature. He's a teenager now, so we don't have to fret so much about him being out in the sun as... he's turned nocturnal again (he was as a newborn, too). But again, as his parents, the decision not to subject him to something his body obviously couldn't/can't tolerate was ours to make - not that of his grandmother.
I really hope your son feels better soon. But as his mother, you're absolutely not being unreasonable in the slightest... and, like others, I'd be very wary about leaving your child(ren) with your MIL in future. We couldn't leave our son with my ex-MIL for years because of her habit of grabbing an arm and dollaping lotions onto it without any warning, so when I say "good luck", I seriously mean it, OP 