I'm mum to 11 cats at the best of times and more when kittens are around. As you can expect with this furry little lot, then vet trips are often needed with vaccinations for babies, check up's and other bits etc....
Mine don't have a problem with getting in carriers as I always have an open carrier in the kitchen which they all like to sleep in from time to time. I do this so as they never have a fear of the carrier. In fact it's often more trouble getting them out when I need to use the carrier.
Perhaps if you have the room, leave it out and open so as the fear is taken away with recognition of fun times..
Hope this helps.