Consultant - unless you've got a legitimate need to reject providing services to them, I tend to think the same, you should have to serve them if they're trying to pay you, or there's a legitimate business need to avoid that client.
Transport company - it's not your job to judge what's being moved. It's your job to move something from A to B. If you want to avoid moving livestock, don't go into the transport business. Should that same vegan be allowed to not teach kids in school because the kids they teach eat ham sandwiches? Should they be allowed to reject someone from banking services just because they own a fur coat?