In my experience, for most artists, the feeling never goes away. Even really amazing published professionals feel this way, and even if you feel accomplished on finishing a piece, it's not unusual to look at it a few months later and to go "ugh, I thought that was a good drawing?"
You can't get rid of it, but you will get to a point where you've made enough good work, been trusted on paid projects by enough people, or just accumulated a decent enough following to at least have faith that other people believe the work you produce IS good work. Sometimes all you will see in your work is inadequacy, but so long as it's good enough for your audience... well, it's good enough!
The worst case scenario here is: "Nobody reads your comic". Owch, that hurts, yeah. It hurts to think about. BUT... let's say nobody's reading, so I ask around like "Hey... I need some feedback on my comic" and I get some, well, it wasn't a waste of time then, was it? Because I've learned things I can apply to a new, better comic from my mistakes on this one. You always have the opportunity to try again with webcomics, and the fastest way to stop worrying about mistakes is to throw yourself in and make all the mistakes and learn from experience. At the end of it. you'll still have had some practice, you'll still have made something and you'll still have learned useful things. If you chicken out and decide not to draw a comic, what do you get out of it? Nothing, is the answer. So my advice? Just go for it!