The first theorem of Section 4 of this paper, mentioned by J.M. in the comments, gives an affirmative answer, citing V.V Makeev, _Inscribed simplices of a convex body_ (in Russian), Ukr. Geom. Sb. 35 (1992), 47-49 = J. Math. Sci. 72 (1994) (4), 3189-3190, MR 95d:52006:
> **Theorem.** Let $K\subset\mathbb{R}^n$ be a convex body. Then $K$ admits an inscribed similar copy of any prescribed simplex.