In the penultimate chapter of Sphere Packings, Lattices and Groups, the authors define a $196884$-dimensional real vector space and a faithful representation of the Monster group on that space.
Now, because we know the degrees of the irreducible representations of the Monster, this representation must necessarily decompose as the direct sum of a trivial 1-dimensional representation and a faithful real $196883$-dimensional representation.
Choose an vector $v$ in that $196883$-dimensional subspace in general position, and normalise it to have unit length. Let $X$ be the orbit of $v$ under the action of the Monster; it follows that $X$ has the same number of elements as the Monster. Moreover, $X$ is a subset of the unit sphere.
Let $P$ be the convex hull of $X$. Then $P$ is a $196883$-dimensional sharply vertex-transitive convex polytope with symmetry group isomorphic to the Monster group.
EDIT: S. Carnahan's answer provides a more elegant construction, taking $v$ to be a point on a line fixed by a Fischer involution instead of a point in general position. The resulting polytope has 97239461142009186000 vertices, which is minimal (as any permutation representation of the Monster has at least this many vertices).