The following result is Theorem 1.1 in:

**P. Maheux, L. Saloff-Coste,**
Analyse sur les boules d'un opérateur sous-elliptique. 
*Math. Ann.* 303 (1995), 713–740. 

> **Theorem.** Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and
> $p^*=np/(n-p)$. Then there are constants $C=C(p)$ and $A=A(p)$ such
> that for every $r>0$ and $\phi\in C^\infty(B(x,r))$ we have $$
 \left(\int_{B(x,r)}|\phi-\phi_{B(x,r)}|^{p^*}\right)^{1/p^*}\leq  A
 e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}}
 \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}, $$ where  $$
 \phi_{B(x,r)}=\frac{1}{Vol_g(B(x,r))}\int_{B(x,r)}\phi. $$

It follows that under the above assumptions, if $\phi\in C_0^\infty(B(x,r))$, then
$$
 \left(\int_{B(x,r)}|\phi|^{p^*}\right)^{1/p^*}\leq  A
 e^{C(1+\sqrt{|k|}r)}r\, Vol_g(B(x,r))^{\frac{1}{p^*}-\frac{1}{p}}
 \left(\int_{B(x,r)}|\nabla\phi|^p\right)^{1/p}. 
$$
(Perhaps with differnet costants $A$ and $C$.)

The key point is that the measure on manifolds with bounded Ricci curvature satisfies so called doubling condition and that such manifolds satisfy the Poincare inequality due to Buser. For more information about how the Poincaré inequality and doubling condition imply the Sobolev inequality in a very general setting see:

**P. Hajłasz, P. Koskela,**  Sobolev met Poincaré. *Mem. Amer. Math. Soc.* 145 (2000), no. 688.

For a related question see; https://mathoverflow.net/q/295683/121665