There’s no shortage of options, though they may have different advantages or drawbacks.
The two main things to consider are how the device will connect to the Internet, and power.
To reach the Internet, you have at least the following options:
WiFi. This is ubiquitous and cheap, nearly everybody has WiFi. You’d need a device that can use it, and the details of your friends WiFi (SSID and key, usually). The most common chip to do that os the ESP32 (though there are other options), and there are plenty of boards and even full devices with a button that can do that, like APR Brother’s AButton or the M5 StickC Plus.
One thing to note, though, is that an ESP32 connected to WiFi will draw quite a bit of current and won’t run long on battery. It needs to use deep sleep and wake-on-pin to achieve decent battery life, but be aware that many devices and boards draw a lot more than their specs say because they only count the ESP32’s deep sleep current and forget the LDO or other chips. Most boards and devices will draw hundreds of microamps (or more), a good one will draw less than 20 (like the TinyPICO). That can dramatically change the battery life of the device.
There’s even the TrigBoard which uses an external chip to actually completely power off the ESP32 and only wake on pin, drawing 1.5 microamps during sleep!
Other short-distance wireless technologies, such as BLE, Zigbee, Thread, etc. These can achieve very low currents, but you do indeed need some sort of gateway. If your friends already have a Zigbee network at home, that could be an option. There are lots of Zigbee wireless switches out there, and you can usually make them call an URL somehow (e.g. via IFTTT). It’s the bridge which determines what happens on button press, so you would need to configure your friend’s bridge.
Philips even had the batteryless Hue Tap at some point (it uses the mechanical power of the actual button press to have enough energy to send the message via Zigbee) but I’m not sure it’s still available (I should have bought more back when they were available!). There may be others.
Note that a Zigbee switch/button + a bridge will be a less than $100.
LoRaWAN. You need to be within coverage of a LoRaWAN gateway, like The Things Network (TTN). Coverage can go for miles outdoors, but is a lot more difficult to predict indoors. In some regions regulatory constraints make it difficult to use confirmed packets, so you could end up with the device in “fire and forget” mode, sending packets but having no idea if it got to its destination. This may or maybe not be an issue for you.
Cellular (LTE-M or NB-IoT). Again, you need coverage, and you’ll have to pay for traffic (though this should be quite cheap) and possibly a small monthly fee, depending on the country and carrier. But this will work anywhere there’s coverage, and no need to configure your friends’ WiFi or anything. Battery life may be an issue.
Until here I have mostly considered that you want to operate the device on battery with no wires. If wires are not an issue, you could simply plug an USB adapter into the device (and then an M5 Atom Lite is probably the cheapest option), or even use PoE.