Is it permissible to perform ablution with irrigation water that is owned by the garden owner (extracted artificially with the contribution of farmers)? And does it have the same ruling as performing tayammum with pebbles and dust if it is in the form of a commodity?

If the water mentioned is pure and does not prevent one from performing ablution with it, then ablution performed with it is valid, whether it is extracted by a machine or flowing. All types of water are suitable for performing ablution. However, if the questioner is referring to the mineral after it has been transported, owned, and sold, then it is not suitable for tayammum according to some scholars, such as Al-Shawkani, who holds that tayammum is only valid with earth from which dust adheres to the hand. Therefore, tayammum is not valid with transported minerals.