If vec is a CVX variable or expression (which must be affine, else norm(vec) wouldn’t be allowed), you can not divide it by norm(vec).
However, there are some situations in which that denominator can be multiplied out. For instance, if x were a scalar CVX variable, then instead of x/norm(x) >= 1, you could enter x >= norm(x). I don’t know whether there are non-trivial cases in which this would be useful for a non-scalar CVX variable.