It isn't illegal.
I personally would still advise against it. I have a lot of piercings myself. Proper piercers will only pierce with a sterile needle (much less painful, much cleaner and much gentler than a gun) and will also only pierce children's ears if the child is old enough to explain themselves that they want it done, and can demonstrate that they understand the aftercare process. The place I go to doesn't pierce the ears of children under the age of eight at all.
There will certainly be jewellers, beauticians etc who will pierce a baby's ears, but they will use a piercing gun which is a lot less hygienic and can also be quite dangerous on a baby's ears - a piercing gun basically uses the earring itself to pierce the ear by forcing it through (a bit like using a stapler). The earring isn't very sharp so it doesn't pierce cleanly like a needle, and the actual force of the gun is quite strong, so can cause bruising and cartilage damage even when it's only a lobe piercing, because a baby's ears are so small.
I do have sympathy with your husband's family because, as you say, there are cultures where ear piercing almost at birth is totally normal and standard and of course, in many cases it goes ahead with no problems. Culture and connections and so on can feel really important to people and to them, there is nothing wrong with this practice. But I personally wouldn't risk it for what is essentially a cosmetic procedure - I think I would just explain that although it's part of their culture, on this particular issue you want to do things the way your own culture does it, and wait until your daughter is older.