Yes, but non-functorially in the compact Lie group $G$.
In 1951 Norman Steenrod produced the very first model [4, Theorem 19.6] of a $n$-universal space [4, Definition 19.2]. See also the equivalence of his [4, Theorem 19.4]. His $E_n O_k = V_k(\mathbb{R}^{n+k})$, a Stiefel manifold.
Strangely, this was not acknowledged by his Princeton colleague John Milnor in his 1956 article [5], which actually has no references! (Was this rush due to the Cold War, though the Space Race started with Sputnik in 1957?)
Namely, since any compact Lie group $G$ is known to embed into some orthogonal group $O_k$, Steenrod's $B_n G := O_{n+k}/(O_n \times G)$ is a closed real-analytic manifold by the classical slice theorem. Hence it's finitely triangulable by opaquely Cairns's thesis [1], first using Whitney's embedding [2, Theorem 1] of $B_n G$ smoothly into some euclidean space.
This triangulation process was elucidated by Whitehead's followup article [3, Theorem 7] and later by Whitney's book [6, Theorem 12A]. (Is this why people nowadays call it the "Whitney triangulation"?)
[1] Stewart Cairns, On the triangulation of regular loci, Annals Math 35(3):579–587, 1934
[2] Hassler Whitney, Differentiable manifolds, Annals Math 37(3):645–680, 1936
[3] John Whitehead, On $C^1$-complexes, Annals Math 41(4):809–824, 1940
[4] Norman Steenrod, The Topology of Fibre Bundles, 1951
[5] John Milnor, Construction of universal bundles II, Annals Math 63(3):430–436, 1956
[6] Hassler Whitney, Geometric Integration Theory, 1957