A full week into classes, many of my students have not yet been able to obtain their books.
This happens at my campus too. It's not likely to change, because it's an inventory management/cash flow strategy for the bookstore. Come October, they don't want to be left holding any unsold books. Those books represent a huge cash outlay, and shipping them back to the publisher costs money and uses labor. Many faculty don't assign any work from the book for the first week, so the bookstore finds it convenient to act as if all classes are that way. Some students are shopping for classes, so the store wants to take their returns and sell them to other students in the second week.
My question is this: As a faculty member, do I have any ability to coerce them to improve their service? For example, if I were to join the relevant Faculty Senate committee, and pester a bunch of people in my university's administration, would I be able to bring about change?
Not likely. Your campus outsourced this function for budgetary and administrative reasons. In particular, if your campus workforce is unionized, then outsourcing the bookstore lets them avoid paying union workers, whose wages and benefits are expensive. Because it's outsourced, they can't directly control the store's operations, and they outsourced precisely because they didn't want to control them.
Although this issue has both educational and management aspects, it's primarily a management issue that happens to impact education. Your faculty senate only deals with professional and educational issues. Its job isn't to get involved in the operation of the campus. Sometimes a faculty senate does succeed with an expansive interpretation of its mandate, but in my experience that gains sufficient momentum only when it's an issue involving wages, benefits, or working conditions, and they can act in concert with a faculty union.
You do have some control over this at the individual level. You can use open educational resources. Although I'm an OER enthusiast and use them exclusively in all my classes, you should not be under the illusion, as suggested by some other answers, that this will cause a prairie fire of resistance that will fix the whole problem. Most faculty want the convenience of the publisher's ancillaries, including the test bank, etc., that they already have set up and are used to. Few faculty care at all about either the price or the didactic quality of texts. Most will express attitudes that these are irrelevant: -- that the students wont buy the book anyway, or won't read it, or will use it only as a supplement to the prof's own (superlative) lectures.
Keep in mind also that that there are likely to be campus financial interests involved. E.g., on my campus, the bookstore has a 37% markup (which they prefer to describe as 27% of the retail price), and any profit goes to sports teams. And although I haven't seen any reliable evidence of direct cash kickbacks to faculty, it is indeed common with the big-bucks freshman survey texts that there will be "soft" kickbacks. E.g., on my campus, the publisher of the freshman calculus text invites faculty on trips to Florida to meet the author.