The use of animals is pretty much unavoidable if you are worried about it from a moral point of view. Many of the people around you will be using animals in their research and it will be hard to avoid, unless you work in human psychology or chemistry or some engineering fields. Most biomedical research involves animals in some way or another. If you just have allergies, that is another thing and direct exposure can pretty much be avoided if you pick the right lab and have competent people around you who are willing to help out. But if you are morally opposed to animal research, it is important to understand that even the antibodies that you will be using are derived from animals. You could also probably work with bacteria or something, but even then you will be using animal derived serums to feed the cells.
In other words: there are ways to avoid direct animal handling, especially in labs with lab techs who do all of that stuff, but if your reasons are moral, be very careful which field of science you choose to go into. If you choose to work in a lab that does animal research, even if you personally start off on a project that doesn't involve animals, there is no guarantee that your project won't end up going in that direction. I imagine that saying no to your PI in that sort of a situation could be VERY unpleasant.
(I realize that the OP is probably just asking out of curiosity, so know that I am writing to any "you" who might be reading with those reasons in mind.)