Is obtaining a professorship contingent upon the ability to secure federal grants? I am a DACA student in the US in my last year of my PhD program, and that means that I will be applying for postdoc and professor positions. However, DACA status means that I am ineligible to apply for federal research grants and I am limited to only applying for private grants that have no restrictions on legal status.

There's also a lot of uncertainty surrounding the fate of DACA as the Trump administration has previously stressed that they would work to rescind the program (which may result in deportation of people like me). 

My advisers said that applying for lecturer positions should not pose a problem, but I'm concerned about my chances to apply for professor positions when I am not able to secure any federal grants issued by the government, in addition to the fact that my stay in the US is not 100% guaranteed in the long run. I love teaching but would also like to continue doing research as well, in a university setting...

Any insights on the matter, especially by those that have worked in hiring committees, would be much appreciated!
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.