I was originally going to distinguish between those influenced by Richard Thompson The Guitarist and Richard Thompson The Songwriter (and I would have thought Richard Thompson The Vocalist would be a harder sell as a major influence on anybody, but Dave Alvin has talked about learning how to sing from him).
Anyway, RT The Guitarist has a style that is one of the most literate in rock music. It incorporates influences from such disparate elements as: The Shadows' surf rock, Charlie Christian and Django Reinhardt's swing jazz, McCoy Tyner's fractured chording, Gene Vincent and James Burton's essential rock & roll sound, the modality of the traditional folk music of the British Isles, the minimalism of Robbie Robertson (and Robertson's Pops Staples fetish), the melodicism of Clarence White and the Byrds, and a healthy dose of the avant-noise of Ornette Coleman. His electric sound is usually pure Fender-on-Fender, the clean Strat sound through a lightly reverbed Fender amp. He prefers the 4th Strat position, blending the middle and bridge pickups into a clean quack. To me, that's the purest electric guitar sound in the world. But he also overdrives his sound when he needs to. One of his main guitars has a neck-position P-90 pickup and I believe his pedals include an Ibanez 808 (or a similar boutique pedal).
RT The Songwriter pulls from many of the same sources - the Byrds, Sun Records, the Band - and produces meticulously constructed songs that span all sorts of rock and folk genres. In his best, there's generally - but not always - a world-weariness in the lyrics, a sense of looking beyond the surface for greater meaning, often an unreliable narrator, and a structure with unassailable surface logic that disguises its unconventionality.
In rock circles, Mark Knopfler is an obvious example of a RT influencee, his entire career built on sounding as much like Richard Thompson as possible. Richard Lloyd of Television has a similar tone, but apparently did not discover Thompson until after recording Television's first two albums. However, Lloyd's brief replacement in Television, Peter Laughner, was a huge fan. Laughner also co-founded Pere Ubu (and David Thomas of Ubu made several albums with Thompson), and I think Thompson's influence on indie rock is considerable. Pete Buck of REM. Bob Mould, with his nutty tunings and fractured chords, is, like Lloyd, a latecomer who sounds like he grew up on Thompson. I hear quite a bit of Thompson in Curt Kirkwood's style on Meat Puppets II and Up On The Sun. J. Mascis of Dinosaur Jr, definitely. Ira Kaplan of Yo La Tengo has internalized Thompson's clean and dirty tones. Lou Reed has talked about how much he loves Thompson, but that would be later Lou. Robert Quine sounded like Thompson sometimes, but I think he and Thompson share a lot of influences.
On the folky-countryish side, Buddy Miller owes the man a clear debt. David Alvin of the Blasters & the Knitters. David Hidalgo of Los Lobos. Any number of semi-adventurous acoustic players, like Bruce Cockburn and David Wilcox (although neither of those guys does anything for me). Michael Hedges used to cover one of his songs live.
John Cale's albums from the 70s sound like he's been listening to Richard Thompson and the Fairport Convention, and Thompson plays on his 1974 album Fear. Elvis Costello covered the man, and it would be surprising if Stiff-era Nick Lowe wasn't a fan. Besides many of those already mentioned, he's been covered by X, the Del McCoury Band, Tony Rice, Mary Lou Lord, the Bis-Quits, Jo-El Sonnier, Beausoliel, Graham Parker, David Byrne, Tortoise & Bonnie "Prince" Billy, Linda Ronstadt, David Gilmour, Bonnie Raitt, the Neville Brothers, Emmylou Harris, Victoria Williams, Arlo Guthrie, Caitlin Cary of Whiskeytown, and Albert Lee (who Thompson would probably consider an influence). The late, lamented Mendoza Line (and the side project Slow Dazzle) specifically mentioned his influence in their liner notes.
All of this is to say that the man casts a long shadow over rock, indie rock, and folk circles.