Philip remarks that he wants to define the Hecke operator as an endomorphism of the Jacobian of "any Riemann surface $X$ such that the endomorphism ring is defined over $\mathbf{Q}$", and at the same time he wants the Hecke operator to reduce to $\{\rm Frob}_p + p{\rm Frob}_p^{-1}$. I think that for a generic Riemann surface $X$, the endomorphism ring is $\mathbf{Z}$. However, $\{\rm Frob}_p + {\rm Ver}_p$ is very unlikely to be an integer, so for a general $X$ it is highly unlikely that there exists an element of ${\rm End}(X)$ that lifts $\{\rm Frob}_p + {\rm Ver}_p$. (I'm using ${\rm Ver}_p$ to denote the dual of Frobenius.)

Here is another closely related point. When $A$ is an abelian variety with good reduction at a prime $p$, there is a natural map ${\rm End}(A) \to {\rm End}(A_{{\mathbf F}_p})$. (See my remark here). I *think* this map is injective (consider the induced map on Tate modules at some good prime $\ell$). Thus you could define the Hecke operator $T_p$ to be the unique (if it exists!) lift of ${\rm Frob}_p + {\rm Ver}_p$. That's intrinsic and makes no reference to any moduli space.