Can I legitimately get any hats for free?

There are two ways of getting hats/cosmetics in TF2 for free.

1. Complete promotional actions (most are already finished or require you to spend money elsewhere)
2. Get one in the drop system

If you're looking for a hat through non-grindy means (waiting to get stuff from the drop system), you can try earning the Spiral Sallet, by earning Mission Accomplished achievement from Spiral Knights, for example is still an on-going "promotion".
Spiral Knights is a free-to-play game.

Alternatively, you can wait for the drop system to give you a hat. Which occurs around every 100-180 drops^.

A reminder that hats earned through promotions and achievements are untradable and unmarketable (unless in Genuine Quality, for most cases which makes them marketable AND tradable), and hats recieved through the item drop system (unless you're using a Free account) are tradable but not marketable.

^This figure is not accurate. It is an estimate from what I've received, and it DEFINITELY is not how the game calculates when you get a dropped hat.

The easiest hat to get for free is the Ghastly Gibus. To get a Ghastly Gibus, dominate a player who is also wearing a gibus to earn the achievement and you will receive one. The gibus can be worn by all classes.

If you don't want to spend any money at all, which means continuing to play the f2p version of TF2, getting a hat from drops is extremely unlikely- most cosmetic items can't drop. To upgrade, you'd need to make a purchase from the Mann Co. store- even if it's just spending 50c on something.

Trade weapons for scrap on scrap.tf really it works well enough.Also 300 hours played on (if I'm correct) a premium account (I don't know if it works for F2P accounts.)