Basic power: drive 12v LED from 5v Raspberry Pi output — no 12v supply available
In general, you need a boost converter on the 5V supply input,  to 12V. You can typically see 85% efficiency. You still want to use the transistor on the RPI output, since it is 3.3v and low current. 

Any typical step up or boost converter will work,  considering you keep in mind the minimum regulating load. 

Alternatively you can find a boost converter with a 3.3v enable pin, and control it directly from the rpi.

Edit The button that OP linked to can be disassembled by design, to replace parts. An easier solution is to replace the led resistor with one suitable for 5V operation. I'd still advise a transistor setup. If you feel lucky punk, well do you?, then a resistor suitable for a red led at 3.3V 16 mA or less would work directly, with minimal brightness change, but a ten cent transistor to protect the RPI pin is recommended.