Is it possible to survive in university academia without applying for grants?
There are certainly some situations in the US where you can be a tenured professor, do some unsupported research, and never bring in grants.  However, the expectation that you will bring in research grants is fairly wide spread, and having at least some grant funding is necessary to obtain tenure at a surprisingly large number of institutions (not just the "Research I" nationally ranked universities.)  

Research funding varies dramatically between disciplines.  In the following I'll restrict my attention to mathematics since you mentioned that was your area.  I'm including statistics since in many cases statisticians get jobs in mathematics departments at these kinds of institutions.

The exceptions boil down to universities in which research is not a high priority and teaching is more important.  These institutions are typically "Regional Comprehensive Universities" often with names of the form "directional state university."  For example, in New Mexico we have "Eastern New Mexico University", "Western New Mexico University", "New Mexico Highlands University", and "Northern New Mexico College"  

Teaching loads at such institutions are typically high (3-3 or 4-4, plus lots of advising) and graduate programs are small or non-existent.  Undergraduate
science and engineering degree programs are also small.  Many of these universities don't offer any degrees in engineering.  Much of the enrollment in math and statistics courses is in freshman level service courses like college algebra and intro to statistics.  

In order to get hired into a tenure track position at such an institution you'll need a PhD and some teaching experience (many candidates for such positions will have worked in "visiting assistant professor" positions for a few years.)  There will typically be some expectation that you will publish research during your time as an assistant professor, and you may be encouraged to apply for research grants, but very few faculty at such institutions actually have grant funding from the NSF or other federal agencies.  

In mathematics, the number of tenure track positions has been in decline in recent years.  The competition for these kinds of positions has become extreme, with institutions reporting hundreds of applications for each position.