**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].

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][0].  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




  [0]: https://en.wikipedia.org/wiki/Slice_theorem_(differential_geometry)