What's the easiest way to get a physical button that I can stick some batteries into, and it'll trigger that HTTP call (over the wifi) when I push the button?

The IoT Button is a versatile smart switch offering dual Home Assistant integrations through ESPHome and Zigbee. Powered by ESP32-C6, it features customizable RGB LED indicators, programmable event triggers, USB-C charging, and flexible mounting options - everything you need for intuitive smart home control.
@foone ooh, this just needs a esp32 worth a button. I just got a thing from M5 for home assistant that has at least that much. They've got a bunch of things similar.
I got this one, but I'll have to look for a better match for what you're wanting
https://shop.m5stack.com/products/atom-echo-smart-speaker-dev-kit
@foone they also have this chonky fella I wanna do something with
https://shop.m5stack.com/products/chain-mechanical-key-button-stm32g031

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@foone easiest? Wireless nes style controller. It's lots of buttons, but you could ignore most of them, or have them all so the same action.
Or those tiny wireless keyboards the size of a wallet
Most of the things that were good for that have vanished from the cheap places.
I should have bought so many more WiFi mains switches when Poundland* was filled with them . . .
*Those were not £1. They were £5 but it wasn't bad for an easily co-opted, mains powered Tuya ESP microcontroller module with a relay to switch mains current, all packaged in a nice inline case to fit any appliance.