By which I mean is it possible to unlock every weapon in the game without ever paying for one, or can some weapons only be obtained through micro-transactions?
Any item which could have an effect on the game can either be found as a drop, unlocked from achievements or be crafted.
The only items which cannot are reskins of certain weapons, some cosmetic items and Strange (kill-counting) Weapons. Most of these are promotional items from other games and all behave the same way as an acquirable weapon.
A full list of items which aren't dropped can be found here: http://wiki.teamfortress.com/wiki/Item_drop_system#Non-dropping_items
A Brief History of TF2 Weapons and Items
Long ago, TF2 Weapons could only be earned through the achievement system - by earning a certain number of achievements for a given class, you would get one special item for that class.
This is no longer necessary - weapons drop at a semi-random, consistent interval. You can find out more about the speecific rate of item drops on the TF2 Wiki, but suffice to say, you will get a fairly substantial number of weapon drops just by playing the game regularly each week. Though it may take some time for the random item drops to give you the exact item you want - you can remedy this by finding players who are willing to trade for the weapon you desire, and you will get duplicate weapons, so you will have some to trade, though you can only do this if you have a paid-copy of TF2 (either bought before the F2P update, or bought an item at the MannCo Store).
While every type of weapon can and will drop, you cannot get items with special qualities through random drops. This includes:
Note, however, that special qualities do not affect weapon performance. Every item in the game, regardless of cosmetics or quality, has the same capability as an item of its kind.
There are other special qualities and items that cannot be found from random drops or without money. And, theoretically, you could save up enough weapons through random drops to craft into enough metal to buy enough rare items to eventually be able to trade for any item in the game. Theoretically. Assuming the item you want is up for trade, and CAN be traded. Theoretically.
Yes, every weapon in TF2 can be found without having to pay any money. The random drop system may take some time to give you the weapons you want though. You also cannot find special quality items, but these do not affect gameplay.
Yes, all gear, including weapons, can be acquired trough either simply playing the game having them drop, trading or crafting. It just requires a boatload of time and some luck.
However, some unique quality weapons might need to be purchased, but "quality" does not define weapon performance, so a "normal" quality weapon is just as good as it's "strange" version. So IF you want all versions of weapons as well, then yes, you likely need to use real cash.
While you can trade for some items, ultimately certain items had to have been paid for at some point. For example, Strange Bacon Grease, randomly obtained by unlocking crate #50 with a key originally purchased for $2.49 from the Mann Co Store, is an item which makes the Frying Pan, acquired by owning Left 4 Dead 2, Strange, meaning it can count kills. For the most part, items that were paid for at some point of their existence are purely cosmetic and make no difference in play from their standard version.
When it comes down to it all weapons except for reskins (such as the Mutated Milk, which comes from a Bread Box, a crate specifically released for the Love and War update) and a select few promotional items, namely the Iron Curtain from Poker Night at the Inventory as well as the other items from that game, and the Sam and Max items, are available to be dropped randomly at no cost to you except for play time, and may also be traded from other players at a low value unless they happen to be newly-released weapons or also crafted using your own weapons and scrap metal.
Additional promotional items include Genuine versions of weapons and hats like the Robo-Sandvich, another reskin which only exists as a Genuine item as far as I know, from toy models that Valve and its affiliates sell through the Valve Store as well as other sites.
I can tell you that at least some weapons need to be paid for if you are only just starting,
All the "Vintage" versions of weapons were only obtained by players before the new content was released, these cannot be unlocked anymore and so have to be purchased.
There might be more weapons than this but this at least answers your question that there are some that must be paid for.