Would it be possible to create sound distortion similar to that in fifties guitar amps with steampunk tech? This is regardless of the resulting amp's size, cost, etc.
The first triode valve was invented in 1906, and first used as an electronic amplifier in 1912. Therefore using valves is not an option.
Prior to 1912, the only way to get a high-level audio signal was a carbon microphone. These were used in old telephones up to the 1970s. The frequency response is poor and the noise level is high.
A guitar amp also needs a loudspeaker. This wasn't invented until 1925. Even if you had a loudspeaker, you would not get enough power from a carbon microphone to drive it.
In short, guitar amps are a 20th-century technology.
What you might be able to get is distortion or harmonics by including resonating strings. Look at the hardingfele (a type of violin) that has the four regular strings and several additional strings that are not bowed, but resonate and hum. The Hurdy-gurdy also has resonating strings.
If you just want the look, coppersteam have made one off pieces. These are intended as general purpose, low distortion amps.
Steampunk is much broader than re-creating electric anything. It's about blazing new trails using (meta-) scientific principles to make life better - or at least more convenient. Snake oil salesmen made out like bandits because nobody (but them) could understand the reasons why their machinations worked (even if they didn't)! When you decide you want to do something in Steampunk, you have to think more about what you want the something to do, rather than how we accomplish it today.
Back in the day, there were all kinds of experiments where someone would use a "speaking trumpet" (when reversed and placed to an ear it improved hearing - what a concept!) as an input device to their constructs - and lots of our current effects simulate different rooms or chambers, so that plays into your theme as well. It won't be an "amp" like what you might expect: it might be something the size of a performance stage, with various HVAC-sized pipes (the sides of those guides would need to be rigid) running between the various stages, filters, and tanks. But you could do it, using small-ish chambers, steam-powered fans to provide modulation and metal plates to provide distortion.
But that's only if you decide to plan to build it today, using 1900's technology. There's something almost "magical" about steampunk, and its ability to come up with "entirely new" methods of accomplishing tasks. (Meta-science! Orgone energy springs immediately to mind, here...) Want the sound/noise to somehow get louder? Your professor-type would tell you that you need a resonance chamber of [x] size to get that much volume, (wrongly) referring to how an auditorium makes your voice louder so everyone can hear your stage whisper. (Remember the times: the descriptions sound hilarious by today's technical standards, yet they seemed reasonable in light of the technologies they were employing back then.) Rotating fans placed within the "waveguide" of the sound would also cause weird effects (go talk into one and see what I mean) and remember that most steam-powered gear is rather silent itself, so noise-abatement from signal to output would be relatively clean. You have a LOT of wiggle-room when putting this thing together. And remember that different metals make different sorts of noises (according to the tech -er- beliefs back then) so you have some more wiggle-room in making it "fit in" for the times.
In short, it won't be portable, as in something you could take to a gig. You might be able to move it from place to place with a team of interns (followers of the Genius Doctor Dynne) and a horse-drawn wagon or two (the second to drag along the tools needed to re-assemble and maintain that monstrosity!) but over time such a device (well, collection of devices piped together) would become quite sophisticated, especially if the mad scientist who built it was disassembling and reassembling it as she traveled from show to show. It's all about filtering, really, as the mutational evolution of the pipe organ illustrates most elegantly. Your "amp" would probably wind up being massive, with a steam-generator (a boiler) located somewhere outside the stage area where you wanted the music to come out, because nobody wants to hear the clunk of firewood (or rattle of coal down a chute) during the quieter performances of whatever was being played.