Scout- Probably the most likely to survive. A good scattergun shot will end a zombie quickly, as would the baseball bat. His speed would also be useful in escaping from zombies.
Soldier- bulky, slow, but deadly. Their rocket launchers will kill zombies easily, but ammo for them is rare. The shotgun is always useful, and the shovel makes a good melee weapon.
Pyro- pretty weak on his own, but effective in teamwork. Setting zombies aflame takes time to kill them, and is cumbersome for sole survival. The axe, however is one of the best decapitators around.
Heavy- The most useless. Minigun ammo would be extremely rare and wasteful as it only takes one shot to kill a zombie. Shotgun is fine, but fists would put one at risk of being bit. Their slowness doesn't help much either.
Demoman- Defensively, pretty useful. Ammo would be rare, and the bottle wouldn't help much. There's also the added risk of blowing yourself up.
Engineer- A defensive mastermind. A few sentries will keep back the Z's, and repair is as simple as finding some scrap metal.
Sniper- Getting distance would be vital for the rifle to be effective. The machete's a good weapon, but the smg is useless.
Spy- uselss. The revolver could get some headshots, but the knife would need headshots. The cloak and disguise wouldnt' work with zombies being able to smell humans.
Medic- Not the best for combat, but regenerating his own health and being able to heal others would be vital for group survival.
I would personally choose scout. Bonk and run, men. Bonk and fucking run.
Edit: Note, I'm using zombie logic from "Zombie Survival Guide" and "World War Z' by Max Brooks, and FVZA.com.
Edit 2: Oh wait, I just read the no ammunition part. In that case, Soldier. Undead go boom boom forever.