Not true, LPRs can own guns, people on non-immigrant visas also can but they basically need a hunting or sporting license to do so, dunno about TPS holders but I'd lean towards yes because TPS is a non-immigrant status.

The relevant law here is the National Firearms Act, which does unfortunately list us as "prohibited persons", so we're basically lumped together with convicted felons and the dishonorably discharged.

I'd love to see a 2A case brought by a DACA recipient since we're not exactly illegal immigrants but I'm sure all the chicken hawks will screech at me for even daring to think out loud here.