I believe the answer is yes. Let's begin by recalling that if one wants to show that a locally compact group $G$ is of type I, it suffices to show that $G$ contains a "large" compact subgroup $K$, in the sense that for every $\pi \in \hat{G}$ and $\sigma \in \hat{K}$, the multiplicity of $\sigma$ in $\pi|_K$ is finite. This is how Harish-Chandra showed that a real reductive group is of type I (take $K$ to be a maximal compact), and also how Bernstein showed that a $p$-adic reductive group is of type I (take $K$ to be a compact open subgroup).

Now let $G$ be a connected reductive group over $\mathbb Q$. Then, away from a finite set $S$ of places (containing $\infty$), $G$ is unramified and has a model over $\mathbb Z_p$. Let's abuse notation and denote this model by $G$. It suffices to show that $G(\mathbb A^S) = \prod'_{p \not\in S} G(\mathbb Q_p)$ is of type I. The desired large $K$ turns out to be $K = \prod_{p \not\in S} G(\mathbb Z_p)$. This assertion essentially appears (without proof) as Theorem 4 in Flath's article in the Corvallis proceedings. The details are spelled out in the appendix to Clozel's article in the IAS/Park City 2002 lecture notes on automorphic forms (MR2331351; a Google Books preview is available here).