DNA is a very stable molecule. You do not need to chill your samples while grinding for a DNA extraction. I've done 1000s at room temperature.
If it's a tough sample (dried leaves, woody tissue, seeds, etc.), it can help to use a porcelain mortar and pestle or a "bead-beater" to really disrupt the tissue.
Liquid Nitrogen helps to disrupt the tissue. It's useful, but if you don't have access to it, you can do it without it. Just make sure you grind very well you're samples to obtain the best yield possible.
It's true that DNA is a relatively stable molecule, and many protocols allow for DNA extraction at room temperature. However, chilling samples during grinding can still be beneficial in some contexts to prevent the degradation of DNA by endogenous nucleases or to preserve other sensitive components that might be analyzed alongside DNA.
Recently I came across this: "Plant tissues are ground (e.g. leaf, shoot, root, approximately 0.1 g) with 1 ml of the lysis buffer (0.5% CTAB, 1% EDTA, 2.5% Tris base and 5% NaCl) in a sterilised mortar and pestle, without liquid nitrogen."
I didn't try cause I use liquid nitrogen, but maybe it's useful for you. I attach bellow the link of the paper.
Article A universal protocol for high-quality DNA and RNA isolation ...