@HikingProd Great topic! I think it's extremely tough, and something that we should all focus on improving.
I think you can break it down into two things: 1)Underpriveledged communities and 2) Limited access.
Having lived in Washington, DC and remote Washington State, I've been in and around two areas that have a great deal of underpriviledged community members. However, both of these locations had ample outdoor access and activities. In DC, we had miles of trails within the city limits, and a few great National Parks just a short ride away via public transportation. That said, I can't recall anyone from our less fortunate parts of the city using those areas to recreate. I think this comes down to perspective and information. Not everyone looks at outdoor spaces the same, and oftentimes when you're concerned about working two jobs, keeping your family safe, etc., you don't look at Rock Creek Park as an area in which to take the family hiking.
When it comes to access, I think it's a matter of affordable transportation. In Washington State, we lived on the outskirts of Olympic National Park and once again, saw very few people in less fortunate situations taking advantage of everything we had in our backyard. To me, this comes down to the relatively high cost of admission ($25 for two days, or $80 for the year) as well as difficult transportation to and from points of interest. When you focus on putting gas in your car to get to and from work, you're probably not going to drive 120 miles round trip to go for a hike. I think this can be solved, at least locally when thinking about National Parks, by providing cheap transportation options to and from the gateway community. I currently live in Flagstaff and know tons of folks who don't make the 50 mile drive north to the Grand Canyon for this exact reason.