"Factories couldn't afford to buy their inputs", solution?

Well, if your literacy is high enough (50%+) then by far the easiest way to make your factories more profitable is to encourage clerks on the state they're in. If that won't work, then all you can really do is pick up the technologies that improve input or output efficiency. Negative tariffs won't work, since you're probably producing all of those goods on your internal market (check the Production tab of that window and you'll see if you're making more than you're using or not), so you're right: you're probably stuck with subsidies.

I suppose if you made so much of the target material that you could influence the price of it on the world market, you could raise your output of it with technologies or factories.

my strategy is spam craftsmen, soldiers, war by jingo, become a gp.


edit: and subsidize factories so you can keep your ind score.

And additionally:
1. pick techs which improve rgo output
2. conquer provinces with required goods, espiecially when they are owned by uniciv (crap techs). Like China
3. include into your SOI these countries

More resources - less problems.

Lumber is going to be rare and expensive until Russia gets a few techs. Produce more steel and fabric to increase the supply and lower the price.

I think subsidising the factories for short term will give you the resources you need, but it will again become problematic after that. Best thing is to try to get provinces which produce wood, or increase labourers and craftsmen working on the production chain for increased productivity. And you can always keep a small extra government stockpile of things you think you'll need, like the clippers in this case, so that you can supply your navy even when you are not producing anything.

A few hints for keeping your factories supplied with resources.

China usually has all the resources needed until late game, and in great quantities, so conquering or declaring war on them to sphere them (peacefully sphering China is impossible, due to huge population) can get you those resources. If you do conquer their provinces, be mindful that turning them into states is almost impossible due to population size, and Chinese nationalist rebels in the late game can tie up a lot of troops if you are to keep the rebellions in control since they get a lot more members.

Other than that, Russia can produce good quantity of wood if they get some economy tech. Germany will keep the iron and steel available for sale. Britain will keep the coal and dye coming, and France and USA will sell enough fruits and grain to keep your factories supplied. By the late game you might want to sphere those Latin American states for rubber and tropical wood, invade and conquer many of the Arabian nations for oil (why does saying this make me feel so funny?), and also get as many colonies in Africa as you can.

Rank matters as well. The higher your rank is, the higher priority you get for buying goods first. The no.1 power (usually Britain) will buy all the goods it needs first, what remains will pass on to the second one to buy and so on, and the low rank minors can only buy the leftovers in the market, if something remains there at all.