How many residues mod p do you need to take to ensure that you can always find some multiple that contains 3 elements within ϵ of each other
Something like $n>p/\epsilon$ should at least suffice (not sure how sharp this estimate is). Here is the argument.
Without loss of generality, assume that $0\in S$. With any element $s\in S\setminus\{0\}$ associate the set $\{-(\epsilon/2)/s,\dotsc,-1,1,\dotsc,(\epsilon/2)/s\}$, division by $s$ being carried in $\mathbb Z_p$. The total number of elements in all these sets is $\epsilon(n-1)$; hence, if $n>p/\epsilon$, then the sets cannot be pairwise disjoint. This means, there are $s_1,s_2\in S\setminus\{0\}$ ($s_1\ne s_2$) and $i_1,i_2\in\{\pm 1,\dotsc,\pm(\epsilon/2)\}$ such that $i_1/s_1=i_2/s_2$. Denoting this common value by $\lambda$, we will have $\lambda\{0,s_1,s_2\}\subset\{\pm 1,\dotsc,\pm(\epsilon/2)\}$, and it remains to apply an appropriate shift.
Notice also that if $n>r_3(p)$ (the largest size of a progression-free set in $[1,p]$, which is known to be $O(p(\ln\ln p)^4/(\ln p)$), then $S$ contains a three-term arithmetic progression. Therefore, $n\sim p(\ln\ln p)^4/(\ln p)$ works for any $\epsilon$, all the way down to $\epsilon=1$.
For simplicity, I'll assume nonstrict inequality: $x,y,z\leq \epsilon$.
For each integer $t\in [1,\epsilon]$, let $T_t := \{ (\lambda,\mu)\in \mathbb{Z}_p^\star\times \mathbb{Z}_p\mid t\in \lambda S+\mu\}$. It is easy to see that $|T_t| = n\varphi(p)$, where $\varphi(p)=|\mathbb{Z}_p^\star|$.
Existence of $(\lambda,\mu)$ such that $\lambda S+\mu$ contains a triple $x,y,z\in [1,\epsilon]$ is equivalent to having $T_x\cap T_y\cap T_z\ne\emptyset$. Absence of such triple means that the sets $\{ T_t \mid t\in[1,\epsilon]\}$ cover each pair $(\lambda,\mu)$ at most twice, implying that $\lfloor\epsilon\rfloor n\varphi(p) \leq 2\varphi(p)p$, i.e., $n\leq \frac{2p}{\lfloor\epsilon\rfloor}$.
Hence, $n>\frac{2p}{\lfloor\epsilon\rfloor}$ is sufficient.
One can in fact pick $n$ that only depends on $\epsilon$ and not on $p$ and obtain a dilation that is $\epsilon$ dense. See [1] and [2]. In particular, Proposition 2.1 in [1] gives an upper bound of $4/\epsilon^2$ for this more challenging task.
[1] Alon, Noga, and Yuval Peres. "Uniform dilations." Geometric and Functional Analysis GAFA 2, no. 1 (1992): 1-28. https://pdfs.semanticscholar.org/0fb9/e7ad4a1cc2523365d7839b79ad4ad13df7bb.pdf
[2] Berend, Daniel, and Yuval Peres. "Asymptotically dense dilations of sets on the circle." Journal of the London Mathematical Society 2, no. 1 (1993): 1-17.