Everyone needs supplies. Surviving with nothing but the skin on your back can be done…for a couple of days. You want to live past that, you’re going to need at least something.
What your character has access to is going to depend on what character you’re writing and what situation they’re in. Training in survivalism has less to do with it than the character’s personality. Someone who takes survival classes every weekend might find themselves short-shifted if the apocalypse happens while they’re at the mall, but the greaser-wannabe who doesn’t even know that ‘survivalism’ is a thing may have a pocketknife at exactly the right time. Or you may have a character who is safety-conscious and keeps an emergency kit in their car, even though they only expected to ever use it during a sudden storm or car accident. Paranoia, forewarning, and personal quirks have far more to do with what a character may have handy than any amount of knowledge they may possess.