See my 1993 thesis posted at hbelabs dot com slash phd slash thesis dot pdf. It has some older references in its bibliography that answer your question in more detail.
In general, for N>4, the answer is no, at least not all solutions of any given polygon. But then again, it depends on what you mean by in "closed form". If you mean by a finite sum of sinusoids (inside a straight-edged polygon), as you suggest, then only the shapes for which a complete set of solutions exists (in closed form) are the equilateral triangle (Lame' 1852) and the rectangle. Other shapes derived from those, like the 30-60-90 triangle, or 45-45-90 triangle, where one has symmetry-related subsets of shapes, can also be solved completely.
The solvable nature of the problem can be established using the reflection property of the eigenfunctions (and related boundary): If they can tile the entire plane continuously, then the eigen-solutions can expressed as a finite sum of sinusoids. The rectangle, and equilateral triangle are the largest shapes where that can be completely accomplished. The hexagon cannot (even though the hexagon itself can tile the plane) because an odd number of reflections bring the hexagon back to the original position.
The hexagon can be pieced together with six equilateral triangles, resulting in only about half of the solutions being knowable in closed form (related to the equilateral triangle solutions, i.e., those with nodes (for the Dirichlet problem) along lines connecting all pairs of opposite vertexes), the other half are not.
The lowest mode eigenvalue (actually k where L=k^2) of the unit-edged regular hexagon with Dirichlet boundary conditions is somewhere between k_lowerbound=2.67494656521 and k_upperbound=2.67494656694, obtained using the point matching method with about 200 matching points. This lowest mode is not knowable in closed form, but it certainly exists (just not knowable as a finite sum of sinusoids or any other common "elementary" functions).
Using a similar method, the lowest eigenvalue of the unit-edged regular pentagon with Dirichlet boundary conditions is somewhere between L_lowerbound=10.99642708284 and L_upperbound=10.99642708503. None of the pentagon solutions are knowable in closed form.
Other shapes that are pieced together from closed-form shapes, like the famous L-shape (three squares), also have only subsets of solutions that are knowable in closed form based on the puzzle pieces.