Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

Yes, this is due to Hill:

Hill, Gregory, On the nilpotent representations of (GL_ n({\mathcal O})), Manuscr. Math. 82, No. 3-4, 293-311 (1994). See especially Corollary 3.2.

This was generalised and extended by Bader and Onn in

Bader, Uri; Onn, Uri, Geometric representations of (\text{GL}(n,R)), cellular Hecke algebras and the embedding problem., J. Pure Appl. Algebra 208, No. 3, 905-922 (2007).

and

Bader, Uri; Onn, Uri, On some geometric representations of (GL_N(\mathfrak{o})), Commun. Algebra 40, No. 9, 3169-3191 (2012).

This may be redundant as a complete answer with references has already been posted. Just in case this is still useful: I think multiplicity one can be proved by the usual Gefland trick: we need to check that endomorphisms of the induced representation is a commutative ring, this follows once we check that the identity automorphism of that algebra is an anti-involution. The endomorphism algebra coincides with the Hecke algebra ${\mathbb{C}}[P(O)\backslash G(O)/P(O)]$ where $P$ is the group of block upper triangular matrices with blocks of sizes $i, \, n-i$. This can also be realized as the space of $G(O)$-invariant functions on pairs of rank $i$ summands in $O^n$. Switching the two elements in the pair induces an anti-involution. Looking at relative positions of two direct summands in $O^n$ one sees that the relative position of a pair $(N,M)$ is the same as the relative position of $(M,N)$, which yields the statement.