Who invented the guitar amp?

Contrary to popular wisdom, Leo Fender did not invent the guitar amp. His amps used basic circuits that could be found in the Radiotron Designer's Handbook, most of which were invented and patented by the Western Electric scientists working at Bell Labs. Vacuum tubes circuits were first used in audio amplifiers and radios rather than in musical instrument amplifiers.
By the time Leo entered the amp field, Rickenbacker, Gibson, National and other manufactuers had done much of the pioneering work.
Fender started making musical instruments amplifiers at his radio shop in Fullerton, California during WWII. Using surplus parts, the first few were custom built for professional musicians. After the war, he started commercial production with he and Doc Kaufman started building K & F lap steel guitars. At that time, many people saw the electric guitar and the amplifier as a single instrument, so Fender saw the two as interconnected.
Leo Fender had an immeasurable impact on the development of the electric guitar, but in his early years he contributed little to the development of the amplifier. It was his conclusion after several years that the limitation of the speakers and Alnicomagnet pickups available to him kept him from obtaining perfect electric guitar sound. After he developed the Telecaster, Stratocaster, and Jazzmaster pickups, he focussed on improving the tone circuits in his amps to make up for inherent pickup and speaker problems.
Tone controls were Leo Fender's foremost contribution to the field of amp design. When Fender amps started sporting treble, bass , middle and presence controls in 1957, the modern guitar amp had more or less arrived. As a bonus, Fender amps were rugged and roadworthy. Starting in '48, Fender built cabinets that were virtually indestructible. To test durability, Fender sent his amps on the road with bands like Bob Wills and his Texas Playboys. These Fenders were knocked around in buses over thousands of miles of highways and endured countless nights in smoky dance halls and bars. Leo saw the abuse his products got from working musicians, and he built his amps to take it.