The title says it all. I want to read a book chapter from a prominent Oxbridge researcher, but I don't want to shell out the cash for the entire textbook in which it's published. Is it rude to contact the professor who wrote the chapter and request an electronic version? It's for my own research purposes.
Before writing to the author, you should try to find the chapter in other ways:
It's a little rude to bother the author to ask for a copy if you could reasonably get one another way. However, if the book is unaffordable for you and you have no other options, then there's nothing wrong with asking the author. I'd phrase it as a question, to avoid sounding too demanding. The key thing to keep in mind is that you're asking for something unusual, just in case the author has an electronic copy they'd be willing to share (but haven't put on their web site). It's also worth including a sentence about things like your lack of library access. You may not get a copy, but it's worth a try.
Rudeness is largely a matter of tone, IMHO. If you express appreciation for the work in question, explain your situation, and ask if the researcher can help you, I think the researcher should appreciate your interest and want to help. Whether or not that's feasible is a separate (and often legal) question, but as for rudeness, there's no intrinsic reason your email has to be bothersome. Be nice, express enthusiasm, offer constructive comments if you have any, maybe try to phrase your problem impersonally so that it's not just about you and what you want (it's also about the author's impact, and how accessible the work is for the interested audience, which should matter), and avoid common faux pas like connotating entitlement, expectation, or violating cultural norms for emotional expression. Pretty much the same issues as you'd consider when asking for anything from a relative stranger. In summary, be polite!
To editorialize a bit, I'll add that textbook prices are sometimes ridiculous, especially given the economic realities of students, and there are far too many barriers to information access already, so on some level, a researcher who isn't especially beholden to the publisher should sympathize and want to support you!
As per @AnonymousMathematician's answer, you might also want to explain (briefly!) what normal alternatives (such as the answer's suggestions) you've tried and why they've failed you, if you decide to go the route of explaining a problem with access that may concern others. Another common faux pas is asking a question that seems to have an obvious solution; one should at least mention that these won't work, and make very sure that they don't before claiming that there's a problem!
Since one answer already mentioned the high price of textbooks, I'll add: I'm assuming you have already looked to obtain a used copy of the book, but without success. If not, though, I'd start there. I've obtained plenty of $100+ textbooks for less than 15 bucks by looking for used copies online.
If that doesn't work though, when you make your plea, there are a few things you could do that might bolster your chances:
1) Don't act as though you are trying to just scrounge a copy of the chapter; write the letter as though you are trying to start a research relationship. After you explain how the author's chapter will help you, offer to keep this person posted on how your research is going. Presumably there's some overlap of interests; otherwise, you wouldn't be after the material.
2) Let the author know that you're not necessarily opposed to the idea of buying the book outright, if you really like the one chapter. In other words, instead of saying something like this (not that you'd use these exact words, but perceptive recipients of solicitations can often read between the lines):
(That need not be a lie, either. Sooner or later, some more-wealthy relative is going to ask you what you want for your birthday. This experience might help you answer that question.)