(Just so you know, my experience is in physics, so not quite your situation.)
I must admit that, while I don’t know of any schools (including top-tier ones) that required research experience when I applied (10 years ago), it really helps quite a bit for several reasons. First, obviously, it shows that you have thought about research some (and since research is what you do in grad school…), and for that reason alone will promote a candidate who has done research over one who hasn’t. Less obviously, it gives you something to write your statement of purpose about (it’s a little hard to talk about what your research interests are if you’ve never done any) and a professor who presumably knows you well enough to write you a good recommendation.
However, all these can be overcome. If you have really good professor recommendations (especially for lab classes), grades, and test scores, in that order, schools will be forgiving of a lack of research. Doing a fair bit of reading around on your own, coupled with talking with grad students/profs, will allow you to write a decent statement of purpose (and, random advice: I cannot stress enough that you need a grad student, at least, to read and critique this!).
I will also say that it depends quite a bit on the field you are interested in. If it’s a very lab-intensive field, they might look a bit more askance in your not having any experience (“how does this Xylophone know that s/he wants to do research?”) but if you are more interested in theoretical/computational chemistry (for which, presumably, you need a whole crapload of grad-level classes before you can even start doing research anyway) it is much less important. (or, at least, so I’m extrapolating from theoretical/experimental physics)
What did you do during your summers during college? This may also play a role. If you were busy working so that you could eat, that’s one thing. If you were sunning on the beach, grad schools are not going to be quite as forgiving of your not doing some sort of research program during that time.
Can you get a lab job this summer? Even a menial lab job will give you an in where you can write about the research that lab does.