Oh absolutely! In fact, doing web series is a wonderful way to build up a screenwriting resume. Issa Rae, who is the creator of the web series Awkward Black Girl, was able to take her work on that series and ended up with a deal at HBO (Insecure). Video Game High School was independently produced by RocketJump and then they ended up with their own show on Hulu.
The key to this path would be patience and putting in the work. Both Issa and RocketJump produced A LOT of content before they ended up moving to larger things but it was totally worth it because the work they did and the audiences they built while doing is are the reasons they were able to get the attention of those larger opportunities.