We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^{2}(\mathbb{R}^d)$ for $d \geq 1$ is locally Hölder continuous at all energies with the same Hölder exponent $0 \lt \alpha \leq 1$ as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential $u \in L_0^\infty (\mathbb{R}^d)$ must be nonnegative and compactly supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle (UCP). We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures