Is obtaining a professorship contingent upon the ability to secure federal grants?
Setting aside all of the questions around DACA, let's focus on the question of how necessary federal grants are to obtaining (and keeping) a position as a professor.

Professorships at research-intensive universities are indeed generally strongly dependent on the perceived ability to bring in funding.  Among other things, without funding it is difficult to support graduate students. Without graduate students, one's research typically progresses much more slowly (excepting perhaps in certain theoretical areas).  Moreover, at some universities successful graduation of Ph.D. students is part of making tenure.  You may be able to do this with private, state, or international funds, of course, in which case you would want to try to demonstrate why this plan is feasible to a hiring committee.

Professorships at teaching-focused universities, on the other hand, are typically much less tightly linked to the ability to get federal funding. While you may face other obstacles there, your primary job would in any case be teaching rather than research, and small amounts of funding from non-federal sources may be much more plausible as an approach.