Yes they can have a carry on bag or case upto 10kg’s just make sure its not bulky as Ryanair are very quick to make you check bags in.
Good luck – I travelled with them when my daugher was small and they were the least helpful airline ever. I had a buggy and was breastfeeding my daughter and even though I was at the top of the queue (and had queued for almost an hour!) because I sat down at the top of the queue to feed my daughter, they made me go to the back of the queue.
Its like being in a herd of cattle.
Hopefully you’ll have a better experience – I always try to fly Aerlingus when I can. They cannot do enough for you, they are so helpful.