Paper is one of the things one should be least worried about.
We source chips from Canada because it's marginally cheaper to source it from there. It's not like the US doesn't have a ton of sources of wood pulp. Canada just has bigger cheaper sources (comparing like for like quality).
We also burn a lot of "less than ideal for paper" chips for energy rather than feed them into a paper mill, also for marginal cost per result reasons.
Wood chips suitable for paper pulp are also hugely elastic in the same way that recycled metal is. Huge volume is either directed into the supply chain or not based on marginal price. The people making every wood product are choosing what do with their waste based on chip prices and energy prices. Even your local tree service is choosing what to chip and where to dump based on economic conditions and balancing act between relative prices.
If you wanna be worried about something be worried about stuff we don't make much of in the US. Super high volume commodity widgets made from metal, all manner of electronics, etc, basically the kind of stuff where our only domestic capacity is super high dollar stuff to serve defense and aerospace.
It's probably true, because I think wood pulp for paper doesn't have rigid requirements on type of wood, or size, or strength. There are also alternative materials if we had to, like hemp. There's probably a dozen others; I wouldn't be surprised if grass trimmings could be turned into toilet paper.
The threat to lumber for building is a much greater concern because there aren't ready and price-comparable alternatives.
As we learned from COVID-19, manufacturers really don't like building new factories or even changing the tooling to support a different kind of product (such as commercial vs household toilet paper) for a short-lived surge in demand.
An antifragile solution would be learning to use a bidet.
I guess these US manufacturers will need to step it up: Kimberly-Clark, Procter & Gamble, and Georgia-Pacific