Should I capitalize the word 'Web' in this sentence?

No, you should not. Even Sir Tim Berners-Lee does not:

1991 T. Berners-Lee WorldWideWeb: Summary in comp.archives (Usenet newsgroup) 9 Aug., The WWW world consists of documents, and links... The web contains documents in many formats.

OED does say "usually with initial capital", and then goes on to say

Originally written with a capital initial, web compounds are now increasingly written with a lower-case w. Since it is difficult to make an objective judgement about the dominant capitalization in particular cases and the evidence is changing too rapidly for such a judgement to be of any lasting value to the reader, the compounds below have been routinely presented with a lower case w irrespective of the quotation evidence.

The patterns of attachment are less inscrutable. The relative dominance of one-word, hyphenated, and two-word forms is in general governed by the status of the second element. Those compounds in which web is combined with a word occurring independently as a common noun are most frequently written as two words (web page, web site). Blends with parts of other words are typically rendered as one word (webliography, webzine). Hyphenation occurs most consistently where the second element is an adjective or is frequently found in compound formations (web-aware, web-based, web-enabled).

You are using web users, a compound "in which web is combined with a word occurring independently as a common noun" — and should therefore not have a capital letter.

OED then goes on to list compounds where web appears as an attributive: web access, web address, web design, web designer, web developer, web publisher, web publishing, web surfer, web surfing.

As Kris points out in his comment, the choice between web and Web is purely a matter of personal preference (or house style) at the moment—though as time goes on, the capitalized version is likely to become less and less and common, and therefore will begin to look more and more old-fashioned.

Andrew Leach's OED citation provides a succinct description of the current status of Web versus web as the short form of the entity formerly (or formally) known as the World Wide Web. At the computer magazine where I work, we shifted from "Web site" to "Website" about four years ago, and from "Website" to "website" about 18 months ago. Nevertheless, as a matter of house style, we continue to capitalize the W in "the Web." I suspect that inertia is as responsible for our current style as anything else is.

Still, there is one oddity at play here: Why is "the web" rapidly becoming standard while "the internet" remains (I believe) far less common than "the Internet"? Neither "Web"/"web" nor "Internet"/"internet" is a proprietary name; and as far as I can see, "the internet" is even less susceptible to ambiguity than "the web." Does anyone have any insight into the different treatment that these two terms appear to be receiving with regard to capitalization?

World Wide Web, or WWW is capitalized, as is Internet, so I capitalize the abbreviation, Web, since it is one of the three capitalized words in the proper compound noun, World Wide Web. I have had no complaints from my Navy customers. Even if someone prefers Web not being capitalized, I can support my choice with logic.