Officially, no. However, sometimes, some communities invite us to participate community tournaments so we form whatever team that can participate in those.
I mean, I'm going to answer this with some cocky common sense. This is "Serious Garry's Mod" last time I checked Garry's Mod isn't an eSports team. However, we have a steam group that has an SGM tag in CSGO. We have many threads about CSGO ranks, and LoL ranks, ect. You can make a thread on the forums for yourself and make an official SGM eSports team. I don't see the staff team needing to host one. We're Garry's Mod focused right now. Now I may have missed it, but I have been here for 3 years, and I've never seen a community invite us to come play tournaments. But I could have missed it. I doubt this happens often. But that's like a group of friends saying to another group of friends "Yo, wanna go play basketball" It's all for fun. The good thing about how we do it now is everyone has a chance to play, regardless of skill, or position on a team. An eSports team is dedicated, and it's hard to get someone dedicated when they aren't being paid. And no, if we had one paying them would not be an option. (unless of course it was for another game, and they would win prizes)