Our bottle of Carlson's for Kids Cod Liver Oil (lemon flavor, by the way) notes you should consult a physician for children under the age of two. I'm not informed enough to know why that would be; interestingly, they sell Baby Drops of 400 IUs of vitamin D. You could also check a bottle of their fish oil to see if that's okay. You'd get the same Omega-3s without the D. But, that probably has the same note.
For those wondering about doses of this version of CLO, the doses are 1/4 teaspoon for ages two to four and 1/2 teaspoon for age four and older. I can not remember why the one for children is different from the one for adults other than taste. For some reason I think there is something else, so I'd check on those doses for any other version of CLO. I see online that the Carlson's Fish Oil has more Omega-3s, but no A & D, which is good for spring and summer.
I do think it's worth asking your child's physician. My son was two when we started, so that never came up.