Good god don't send her a Mother's Day card thanking her for sharing the children! Those children only exist because of her daughter, she's been there their whole lives, whereas you came later and could leave at any time and they'd never see you again.
She's the one that will be able to tell the funny stories of their mother's childhood, show them photos of their mother as a child, keep that family connection going.
The poor woman is grieving her daughter and sees you in her daughter's place, of course it's awkward. Perhaps she's worried you and her father might try to sideline her.
If she pays school fees she has every right to attend parents' evenings, it's the least you can do to graciously accept her being there.
Remember you're not their parent, but she's their blood grandparent and a link to their mother. Over time hopefully you can show her that you're up to the job, but you can't force it, let it happen gradually by showing humility and that you aren't trying to replace her daughter.
You say her daughter died several years ago, that sounds like it's not that long. How awful and hard it must be for her to see in that short time you've met and married her son in law, and now you seem to want to control the terms of her relationship with her family.
You deal with this by stepping back, being respectful and pleasant to her, do not suggest outings with her or anything that gives the impression you're trying to ingratiate yourself with her.
It's going to take her a long time, if she ever does, to accept you doing all the things her daughter should be doing. Imagine her pain, if you can. You got a shiny new husband and stepchildren, she's got the grief of losing her child, something which no parent ever expects to do. She probably also has negative feelings towards her son in law too, for moving on quickly. Have some sympathy for her and don't try to force or change anything.