Not exactly like this but I did get something similar for my mom after surgery, was a push button control, an extension of the regular cord-believe its real use is for Xmas light turn on and off. Got it at Home Depot.

We have a remote control for our Christmas tree lights. The adapter plugs into the wall, the Christmas lights are plugged in to the adapter and the remote control turns on/off the lights. Should work for any lamp. But not what you want. So......

I think some desk lights have bush button or turn knobs on their bases.

Commercial lamps for hospitality (hotels and motels) are the most common style of lamps with a switch on the base.

Pal said exactly what I was thinking. I always see this sort of switch in Hotel rooms. I wonder if you could identify and tap into the commercial supply sources for the hospitality industry?

A quick search turned up the following supplier. In addition to switches in the base, it looks like many also have outlets (in a hotel this would let you plug in your phone charger or laptop), and there may be Ethernet ports as well. Hard to tell, as the descriptions weren't good.

Here is a link that might be useful: American Hotel Register Company

I have a few with switches on the bases or on the cord. They are all older lamps. I bought one recently at Home Goods for a guest room at our vacation house. It's white and modern, which might be more appropriate for your toddler.

Lamp 2

Lampp 3

Last one...lamp 4

These are all Stiffel and/or Baldwin lamps. If the style appeals to you check eBay.

How do you post multiple pics in a post?

Or you could just purchase the 'Clapper' and not have to touch the lamp at all.

Here is a link that might be useful: Clapper

An electrician could modify an existing lamp for you. You drill a hole in lamp for the switch to come through, and wire the switch into the lamp wiring. There needs to be room in the lamp for the backpiece of the switch.