Although it was once funded entirely by the government, the

I think that all of the critical issues have been mentioned, but as catfreak suggests, we don't yet have a neat account of how an expert test-taker might approach this question. So let me try. Just so that you don't have to go back to page one, here's the text again:

Although it was once funded entirely by the government, the Victoria and Albert Museum was one of the first of Britain’s national museums seeking support from corporations and private donors and to increase income by increasing attendance.

(A) one of the first of Britain’s national museums seeking support from
(B) one of Britain’s first national museums seeking support of
(C) among Britain’s first national museums to seek support of
(D) among the first of Britain’s national museums to seek support from
(E) among Britain’s first national museums that have sought the support of

The Manhattan GMAT approach relies heavily on spotting differences among the answer choices, we ask our students to "scan vertically, and look for splits." But some signals reliably indicate particular grammatical problems, even before those problems show up in the splits. Chief among these signals are parallel markers such as and, but, and or.

When we see parallel markers in an SC question, even outside of the underlined section, we should ask what elements are meant to be joined by them, that is, what elements are logically comparable. Then we should eliminate answers in which those logically comparable elements are not structurally similar.

In this sentence, the first and joins corporations and private donors. That's fine, and since neither of those elements is underlined, there's no issue here. The second and joins the infinitive to increase income... to some earlier element. The only logically comparable earlier element is seeking support..., so that element must also be an infinitive, to seek support.... Eliminate A, B, and E.

That leaves just C and D. Before we go into their differences I want to acknowledge that even a very good test-taker might well rely primarily on her ear to make this choice, because the rules we'll need to invoke are subtle, but C may just sound much worse to some good test-takers.

There are two difference between C and D: (1) among Britain’s first national museums vs among the first of Britain’s national museums, (2) to seek support of vs to seek support from. Both of these splits favor D over C.

(1) Among Britain’s first national museums vs among the first of Britain’s national museums: The GMAT often prefers a phrase such as stone wall (noun-adjective + noun) to a phrase such as wall of stone (noun + prepositional phrase), but the present sentence falls under a rule MGMAT calls "Keep the Prepositional Phrase if You Need To." Catchy name, huh? If you have our SC Strategy Guide, check out page 211 for more info. This sentence isn't really a paradigm example of any standard reason to keep the preposition. My main reason to prefer D here is that we know immediately what it means. Since of Britain's national museums is merely a modifier here, we know right away that the Victoria and Albert Museum was among the first to do something, even before we read what that something was. In C, by contrast, it appears as though The Victoria and Albert Museum is simply among Britain’s first national museums , at least until we reach the infinitive.

(2) (2) to seek support of vs to seek support from : mundasingh123 writes that somewhere in the WWW he found the claim that support of is followed by the thing being supported, while support from is followed by the source of support. He's right. Or the web is right, I guess. There's a little more to it than that, but I don't know that you need that little bit more for the GMAT. Anyway, here goes. Sentences of these forms are all OK:
(A) support from+supporters
The proposed contract won support from both the union leadership and the board of directors.
(B)the support of+supporters
The proposed contract won the support of both the union leadership and the board of directors. This structure require the article the.
(C)in support of+supported
The union president spoke in support of the proposed contract.

I try not to worry about idioms that haven't been tested before. Is this an actual GMAT question? It certainly feels like one, but I didn't see any note about its provenance.