So, unfortunately, microphones don't work as they would in an ideal world. Ideally, the sound you pick up would already sound perfect, but if you're using a bad webcam microphone, the vocals are just not going to sound good. So much of the actual sound comes from 1) the recording equipment 2) the mixing and 3) the headphones/speakers. Without a decent amount of the three, no matter how good or bad you sound, you will most likely sound bad or worse than if you had proper recording technology, a professional mix your track and add the right amount of plug-in manipulation, and a good playback system.
To answer your question: yes, many good sounding artists will sound very shitty on a low-budget system without any mixing.
My recommendations: look into Yeti brand microphones. They're not bottom of the barrel and far from the best sounding microphones but they're extremely versatile and well worth the $100-$150 they cost. Also, I would look into a DAW, nothing fancy or expensive, just to get some basic mixing going. If you have a Mac, Garageband is free and is more than enough. You should look into some quick tutorials on compression, EQ, and reverb.
Also, look into this course:
https://www.edx.org/course/music-production-vocal-recording-berkleex-bmpr365x-0
It's a
free course and covers the basics of about everything from how to use the different microphones, to mixing and whatnot. The guy teaching has worked with Madonna, Usher, P-Diddy, etc. I took this class and learned SO many things.
Feel free to message me with any questions! I'm more than happy to help out a fellow musician!