Non-trad here. I did not have a work permit until I got my greencard. All my experiences - clinical, non-clinical, research were volunteering positions. For clinical and non clinical volunteering, as others have suggested, hospital and soup kitchen are the best places to gain some good experience. Majority of the hospitals usually are welcoming. For research (benchwork basic science research), I emailed every possible PI in universities at a commutable distance. Once I received my work permit, I got paid positions as a CRC and later as a research tech. While I was doing all this, I was enrolled in DIY post-bacc. So, according to your definition I was "outside" the system and not part of any university per se.
You could also take an online hobby class at a community college (check if your employer gives tuition reimbursement) and be considered "inside" the system.