Can this be written in standard "vector calculus notation"? A formula for the gradient of the magnitude of a vector field $\mathbf{f}(x, y, z)$ is:
$$\nabla \|\mathbf{f}\| = \left(\frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial x}, \frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial y}, \frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial z}\right)$$
Can this be written in a form that doesn't mention coordinates explicitly? (i.e., using nablas, dot products, etc.)

I haven't checked if your formula is correct but if it is, then it would appear in terms of the induced tangent map which can be denoted $df$ (in coordinates this would be the Jacobian matrix) as $$ \nabla\|f\|=\frac{1}{\|f\|} df(f).$$
