I mean it is legit. It's been a while since the last time I looked into it. But there are government set-asides for studies like this. I mean don't get me wrong there is still a process of approval and paper work you have to fill out to get money to fund a scientific study like this. But like I said, it's been so long I can't remember what department you have to go through. But there are or have been plenty of scientific studies, surveys and such that they government has funded with tax payer money that some of these studies would be considered wasteful or useful spending. So technically this would make an interesting study. But whatever the guidelines the government lines out, I am sure somewhere in here you would have to include demographics e.g. classes of people, lifestyles may contribute to types of aromas. Like for example, if someone has a poor diet that consists of nothing but fast food (processed) food, would their baby smell different than someone who is a vegetarian, compared to someone who strictly eats organic off the grid type diet. In that case I'd like to know do native American babies smell more like petrichor, opposed to native Alaskan (Eskimos) babies that may as well smell like petrichor, but are the aromas the same, one maybe sweeter, the other more earthy or pungent.
This is a very good study. But you may have a tough time trying to find women who would let a stranger in the room while giving birth to collect samples. Some women you may not. Depending on how in depth and thorough you want your study to comb over. I mean I believe you would also have to consider the at birth aroma and then the post birth aroma, (a day or few days old aroma). And that's just it, I believe in the process of applying for government grants for studies such as this you will have to outline all of this as part of your synopsis on the project.
And to go even further, other countries also contribute monies to studies like this. It's basic human curiosity. And this is the exact foundation where many many great sciences have been discovered inadvertently which have created medicines, products, or simply became a catalyst for discovering other things.