Indeed, you cannot transport resources/supplies through a puppet. Your troops in the puppet's territory will be supplied from the puppet's supply network. The amount of supplies used up by your troops will be transferred from your stockpile to your puppet's the next day.
If you take territory, attacking from a puppet's territory, you get ownership. If your puppet's forces take territory, it's also yours. That said, you'll only be able to hold on to this territory if you can supply your troops, or those of your puppet, in this newly gained territory. You have maximum 30 days to bring in supplies, after that your troops will be out of supply, unable to move into hostile territory, able to move only very slowly, or not at all (in case the unit needs fuel).
The supply system in HOI3 is all about having enough bandwidth to send supplies from the capital / port to the troops. In case you're fighting anywhere but western Europe, you may need to improve infrastructure along the way, or deploy lighter units, or both. The Eastern front is especially vulnerable due to the massive number of troops both sides have to deploy, the poor infrastructure in Russia, and the vast distance to Berlin/Moscow. There is a transfer cost, for each province a packet of supplies/fuel goes through a small amount is 'used up', so the further you get form your supply base, the more supplies/fuel you need to pump into the system (production), and the better your infrastructure needs to be, especially close to the point of origin (capital/port). Seaborne supply is significantly more efficient, and setting up convoys from as close to your capital as possible, to deliver supplies as close to the front as possible, even if the territory is contiguous, can help alleviate shortages, and save on production.