Agree with the above, though you may also want to look for a local lumber store as opposed to an HD or other big hardware department store; they’re more used to accomodating cuts like that. I buy from one in the DC area (Colonial Hardwoods, just in case you’re also nearby) and one sheet of the 5’ by 5’ baltic birch gets me a BUNCH of 12 by 20s, and thus costs very little per 'forge sheet. They actually even ship it to me via UPS (I’m public transportation-bound). One piece of advice – all 1/8" Birch is DEFINITELY not created equal. I bought their other, cheaper ply once and it was total crap. Knot-filled so it didn’t cut evenly, crumbled to the touch – it was clearly meant to be a non-load-bearing backing to cheap furniture, not a laser piece. So definitely test a small order before you place a bulk one.
Alternatively, if you have a table saw, it might not be as much time ripping and cutting as you think, since it’s thin enough to do sheets stacked on top of each other. I had some Baltic Birch left over from when I used to use a machine that took 18 by 24 sheets, and I was able to cut them down just using a circular saw and cutting 5 sheets at once (1/8" thick).
As for cardboard, you can find it cheaply at ULine, they even discount for bulk orders, but I would definitely echo the concerns about flammability. I’ve used it for a few projects or drafts, but you definitely have to watch it like a hawk, and small enough pieces just vaporize – if it’s about the size of a pencil eraser, the laser cuts their outline and the inside just burns away. Uline also offers chipboard, and it’s cheap, but I have no idea how it would cut.