I have a dsi with a copy of NICILAS port of Cave story and im wondering if i need a way screen record to submit runs or is it okay if i just record my screen with my phone

You can use your phone to record it, just make sure that the footage is clear enough that we can see what's going on.

dump your rom then run it in melonDS or bizhawk dump biosi7+9+firmwarei+nand as well if it lag then just TAS it