I'm an adjunct prof in a science deptartment at a community college (I don't teach chem, so no - I'm not the OP's teacher)
Some really good points have been made:
1. If she wants to go into nursing, she NEEDS chemistry - not just for the grade, but for the material. If she's not learning it well from this prof, she should drop it and try again later.
2. from the college's perspective, dropping it "quietly" vs. talking to everyone about it first makes absolutely no difference. Once she fills out the forms, the registrar will process it. End of story.
3. I would, though, encourage her to talk to someone in the science department. Not in a "this teacher is crappy" manner, but approach them saying "I'm dropping chemistry this semester. What can I do now, and next time I take it, to make sure I'm more successful?" My school offers a chemistry 001 class the second half of every semester. It's a non-credit class for students who dropped out of the intro chem class (there are always a LOT of them every semester.) The class goes over some of the basics, including the math skills, that are necessary to succeed. It also stresses a lot of the study skills that students often lack heading into college and introduces the kids to all of the resources available for help.
4. Whenever she takes chemistry, make sure she knows when the tutoring center hours are, when her profs office hours are, what online tutoring is available, etc., etc. I 99% guarantee you that those things are all available to her - for free - at her college. That doesn't mean she'll get a private tutor from the college, but that there are LOTS of resources available for help. She may need to use every single one of them.
5. Finally, I know you're her mom and that you want to help her. And advising and offering our opinions is what parents do. It's in our very DNA. But now's the time to step back and let her handle this. Listen, commiserate, ask questions, but let her figure out what she wants to do and how best to do it.
Good luck!
Thanks! Lots of great perspective in this.