  We study Hecke operators on vector-valued modular forms for the Weil
representation $\rho_L$ of a lattice $L$. We first construct Hecke operators
$\mathcal{T}_r$ that map vector-valued modular forms of type $\rho_L$ into
vector-valued modular forms of type $\rho_{L(r)}$, where $L(r)$ is the lattice
$L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by
lifting standard Hecke operators for scalar-valued modular forms using Siegel
theta functions. The components of the vector-valued Hecke operators
$\mathcal{T}_r$ have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as
generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau
threefolds. We study algebraic relations satisfied by the Hecke operators
$\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer
$n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new
Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of
type $\rho_L$ into vector-valued modular forms of the same type. We study
algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, and compare
our operators with the alternative construction of Bruinier-Stein [Math. Z. 264
(2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].
