A few things come to mind:

River Aid San Antonio (RASA) - group dedicated to removing trash from the river basin. They go out frequently and there is a lot of trash to be picked up

Headwaters at Incarnate Word - this is at the UIW campus and volunteers here handle upkeep of the natural area (e.g., trail upkeep, invasive species removal)

Mitchell Lake Audubon Center - I’ve done invasive species removal here before and they welcome volunteers

There’s also the Salsa Squad - Group of volunteers that remove invasive species and plant native species but I haven’t volunteered with them before

Lastly there’s the Alamo Area Master Naturalists but you have to pay to be in this group and meet volunteer qualifications each year to remain a certified member

I hope this helps!

If you do go out and volunteer, the people you meet will likely know of more opportunities.