Could House Tyrell create a Land Bank of Highgarden even without the proposed Mander-Blackwater Rush Canal? Would the Tyrell’s be able to do that on their own since it would just be within the Reach, or would they need the King’s permission?

Good question!

Yes, absolutely the Land Bank could be built without the canal, it’s just that the two are complementary, with the Bank providing credit for the canal and canal revenue providing a steady stream of capital for the bank.

Most likely, they’d need permission just as the Lannisters probably would. 

Now, it’s possible that if the Land Bank stuck to just being a sub-treasury system which didn’t issue loans but only IOUs for crops that were only good in the Reach, they might be able to get away without needing royal permission unless some bright spark in the Master of Coin’s office realized that those IOUs were effectively money.