Sonichu is to webcomics what 'my immortal' is to fanfiction 
I grew up on independent 70's/80's comics like Elfquest and Cerebus the Aardvark.
They helped shift away from the unrelenting superhero comics that dominated everything.
And, of course Robert Crumb and the Underground comic crowd. They were ... so incredible effed up and talented.
Google Fabulous Furry Freak Brothers if you need a refence.
I was looking more specifically at influential online web based comics but your references led me to the tag of underground comix which just opened up a whole new world to me so thanks
Megatokyo. It first debut in 2000. While I will say it hasn't aged well it did introduce me to the idea that it is possible to use the internet to publish your comics.
Influential Webcomics to me include:
Megatokyo, Penny Arcade, 8 Bit Theater, Exploitation Now, Ctrl Alt Del - heavyweights of early webcomic culture.
Homestuck - One of the biggest kickstarters ever (the biggest at the time) and had a peak readership of millions. Hugely influential on things like Undertale.
Questionable Content, Scary Go-Round - Interesting because of their relationship with the 00s indie music scene and interaction with Bryan Lee O'Malley's early work. Giant Days, which is a Scary Go Round spinoff, is a pretty popular BOOM YA title.
Gunnerkrigg Court - One of the longest continually running story-focused comics around.
Hark! A Vagrant! - You only have to see how many memes this comic spawned to get a feel for what a huge deal it was in the 2010s.
Okay, I'm dating myself, but the ones I religiously read in high school were:
Penny Arcade
Scary Go Round (John Allison has gone on to create several more series since then)
Something Positive (the art wasn't great but the writing was hilarious)
In more recent years I've irregularly checked in to read Hyperbole and a Half (which is kind of a hybrid of writing and webcomic) and often find myself laughing to the point of crying, which is a good sign.
Crap I Drew on my Lunch Break (which I found has also been uploaded to Tapas)
Looking for Group / Least I Could Do
There were a few in the early 2000s that I remember the plot to but not the name. One was a BL had animals who could take human form. The two mains were a snake boy and a half blue, half white gecko boy.
The 1st things that come to mind are Homestuck and Mario Adventures: Playtime's Over. Although I'm not entirely sure if the latter was that big. I just remember Homestuck fan art being all over DA.
My adventure with webcomics started on the Snafu website, specifically with Bleedman's Powerpuff Girls D. It was very popular at the time and the series is still ongoing, but slowly because the author has other side projects.
The rest of the series I followed were also popular but part of the BL branch - when it wasn't this much popular - : Starfighter (I even bought the physical copy during the author's birthday, as a gift) and Teahouse (I think the original website doesn't exist anymore)
A couple of people have already mentioned Girl Genius and Homestuck, but aside from those..
Dr. McNinja ran for thirteen or so years and was pretty popular, though some of the earlier chapters might not have aged super well. Order of the Stick probably wasn't the first big D&D comic, but it was the biggest I knew of at the time. 8-Bit Theatre was also pretty big back when sprite comics were more in vogue.
In terms of gag-a-day comics, can't forget xkcd, which has been running since 2005 and is probably still the biggest math/science/programming comic. Saturday Morning Breakfast Cereal is also a big one.
Achewood ! I was just rereading it before I saw this thread. Probably a influence on my sense of humor
Gunnerkrig Court, vc cats, Emergency Exit, Crow Feathers, Alien Dice.... and more I can't recall the name off
The first which springs to mind for me is Lackadaisy Cats. I stumbled across it on DeviantArt over a decade ago and was blown away by the quality, the tone, the characters, and how funny it was. It was (and still is) a case of an artist putting her whole soul into a comic, and producing a single work of absurd quality.
https://www.lackadaisycats.com2
@crowstories
I completely forgot about Hyperbole and a Half! I immediately clicked over there. I'm so glad she's back, and what a hilarious story to start off the day with. 
She's great when you're in A Mood and want a good cackle. The Sneaky Hate Spiral is particularly good for that.
Gunshow, Dinosaur Comics, Chainsawsuit, Diesel Sweeties, Girls With Slingshots, Saturday Morning Breakfast Cereal, Perry Bible Fellowship...
Definitely Dinosaur Comics is one of those examples that really exemplifies old webcomics the best. It's a strip that's extremely nerdy, focused on one gimmick stretched to the absolute limit, and it somehow launched the creator to becoming a prominent ad company man then later one of the top writers in traditional comics as well.
I also think Gunshow's Anime Club3 is one of the best pieces of media to represent the late 00s/early 10s era internet culture. It's nothing at all that could be famous if you published it in 2021 but back in the day it really worked, I think.
Fun question! Definitely reminded me of some old favorites and brought back memories!
Ones I knew about:
CTR ALT DELETE
Sonichu
Questionable Content (the creator is a local!)
Penny Arcade
The Lease I Could Do
Girls With Slingshots
Ones I read: (haven't kept up over the years but these were a good time!)
Girl Genius
The Devils Panties
Dr McNinja
Red String (This one was kind of THE "girly" romance/shoujo back in a time when they actually were not that popular or common online! Now sites like this and Webtoon are dominated by stories like this. I remember just adoring the art.)
Dead Winter (I think it was/is called?) (I think it was the premier zombie comic that heralded the undead obsession)
Damn, you have me feeling nostalgic for the early/mid 2000's now!
Sonichu has, and will always be more recognizable than any of the premium comics on tapas and webtoons
