My question is, in order to completely model our universe's (3+1)-dimensional spacetime as a manifold in a higher-dimension, flat spacetime, how many dimensions are needed? Will a finite number of dimensions suffice, or are an infinite number of flat dimensions required to completely embed all solutions to the Einstein Equation?

If anyone can refer me to texts or publications that treat GR in this manner (as a curved manifold in higher-d flat spacetime) it would be much appreciated.

That's very interesting. I notice that the Nash embedding theorem (linked from the Whitney embedding theorem Wikipedia page) mentions "isometric embedding", i.e. preserving the length of curves in the manifold, whereas the Whitney theorem does not. Going by this bit of the Nash theorem's explanation:

It looks like a 4-d curved spacetime can be isometrically embedded in a flat spacetime of 39 dimensions or less (m=4 -> m^{2}+5m+3 = 16+20+3 = 39). Unwieldy perhaps, but at least it's a finite number of dimensions.

If the metric for the higher dimensional pseudo-Riemannian manifold is required to restrict down to the metric for 4-dimensional spacetime, then it could take a lot of dimensions.

Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428

Right. In the spacetime manifold in which string theory lives, the extra spacetime dimensions are compactified, so they're not easily physically visible.

A d-dimensional (curved) space(time) can be be embedded (immersed) in a flat space of D = (1/2)d(d+1) dimensions. See L. P. Eisenhart, Riemannian Geometry, Princeton, 1997 (org. 1925), Ch. V, Par. 56, p. 188 or T. Levi-Civita, The Absolute Differential Calculus, Dover 1977 (org. 1926), Ch. V, Sec. 21, p. 122.

Notes:
Flat = it is possible to introduce coordinates x_{m} such that ds^{2} = h_{mn}dx^{m}(x)dx^{n}(x) correctly describes distance (ds)^{2} with h_{mn} fixed (independent of x).
Embedded = When x and dx are inside the sub-space, the ds^{2} above correctly describes the local distances in the sub-space.
In some cases, a flat space dimension < D suffices. Obviously an already flat space of d dimensions can be embedded in d dimensions.
Eisenhart gives an example of the Schwarzschild spherical space time curved 4-d space being embeddable in a 6 dimensional flat space (a special case).
For d = 4, D = 10 is the general requirement.
This only considers general relativity (gravitation), no other forces.

The dimensionality requirement of (1/2)d(d+1) I mentioned earlier was proven by power series development methods (see references Janet, 1926 and Cartan, 1927 in Eisenhart's book mentioned earlier, also mentioned in the ref. below), and hence are local results (hold in some finite open set around a given point) and require that the original intrinsic space(time) metric is analytic in the coordinates. J. Nash, The Imbedding Problem for Riemannian Manifolds, Ann. of Math., Vol. 63, No. 1, Jan. 1956, pp. 20-63, gives a nice historical introduction to the problem and considers cases when less assumptions are made.
http://abdullahburhanbahce.googlepages.com/theimbeddingproblemforriemmannianman.pdf" [Broken]