I work in the regulation of medical devices (like contact lenses). The law varies by state, but you generally have to have a prescription that's been issued in the last 1 to 2 years for a vendor to legally sell you lenses online. That means vendors who are willing to sell you lenses without a current prescription might be willing to sell you counterfeit lenses. And by counterfeit, I mean lenses that might seem to work perfectly well but which haven't been manufactured in sterile, inspected facilities. With no inspection regime or record, counterfeit device and drug manufacturers very truly sell shit all the time that has undisclosed drug ingredients, contaminants, unsafe materials, and on and on. Here's a recent incident that might give you pause about ordering from unscrupulous vendors.
Seriously, just take your kid with you to the doctor. You're not the first person to do so (I have three kids, and when they were young and hard to manage I came to love offices where staff immediately swoop in and are like, oh hey you look like you could use some help, want me to hold these two while you get checked out?).