I think $X:=S^2$ being a compact Riemannian manifold already gives you quite a lot, without any extra structure. The key seems to be the existence of geodesic normal coordinates, and in particular the Taylor expansion they give us for the metric tensor.

Leaving the details of how to do this to one side to begin with, let us suppose that we have constructed a collection of `nice' local parameterisations $(x_i:U_i\to X_i)_{i\in I}$ such that the $(X_i)_{i\in I}$ is an open cover for $X$ and each $H^1_0(X_i)$ is isomorphic as a Hilbert space to the corresponding $H^1_0(U_i)$. Using compactness to pass to a finite subcollection if necessary, taking a partition of unity subordinate to $(X_i)_{i\in I}$ gives us a topological embedding

$$
H^1(X) \to \prod_{i\in I}H_0^1(X_i)\cong \prod_{i\in I}H_0^1(U_i)
$$

into a finite product of normal $H^1$ spaces on $\mathbb{R}^2$. At this point we can either use a diagonal sequence argument taking the compactness of the $H^1(U_i)\to L^2(U_i)$ for granted, or use the embedding to transfer local estimates back to $H^1(X)$ from each $H^1_0(U_i)$ (I think one $\epsilon, C(\epsilon)$-style interpolation inequality plus Freidrich's lemma is enough).

Of course, all this was based on the presumption of access to nice local parameterisations $(x_i:U_i\to X_I)_{i\in I}$. This is where I think geodesic normal coordinates come in. Pulling back the volume form on $X$ using a local parameterisation $x:U\to X$ always gives us a measure on $U$ and an identification between $L^2(x(U))$ and some weighted $L^2$ space $L^2(U,w)$ (where $w$ is the associated density w.r.t. Lebesgue measure). What I think geodesic normal coordinates (probably) let you do is make sure that this density is bounded away from zero, at which point $L^2(U)\cong L^2(U,w)$ and $H^1(x(U)) \cong H^1(U)$.

In case this isn't already clear, I'm a little sketchy on the details of this last bit: the MO post Volume of geodesic balls seems relevant here.

**Actually:** it looks like Hebey's book *Nonlinear analysis on manifolds* has all this and more. I've exhausted my google books preview allowance reading it this morning, but it looks like it covers everything I've said in spades. It also has the advantage of being much more citeable than an anonymous MO post.