Is the end of April too late to find placement in Ph.D programs in the U.S.? I am wondering if it would be worth emailing departments that I haven't applied to this year, asking if they'd be willing to take a look at my application material. Any advice, or should I just wait till next year?

I applied to too few places this year, because I couldn't afford all the application fees. I'm saving up now but I feel bad that I don't have anything planned for next year, and I really want to begin my higher education since I feel strongly about it.
For the most part, yes, it is too late to be considered for graduate admissions at most programs for this coming fall, as most schools have already made their decisions as to who they will admit. 

That said, if you are an exceptionally qualified candidate, and unusual circumstances have taken place, it may be possible to be considered by specific programs. However, there has to be some really strange events taking place. (For instance, one school I know had a banquet during their one prospective students' weekend, and accidentally food poisoned everybody—and ended up with 20% of their expected yield of students. They probably would have loved additional late applicants!)

But, basically, you'd need a department that had a major shortfall of accepted students compared to their research needs, and you'd need to be a candidate that would be strong enough to accept in the regular admissions cycle.