All of the previous answers on this are correct. If you are capable of using the transmuter (it has been in play since the beginning of your most recent turn and it isn't "summoning sick" anymore), and you have the mana to pay for the cost, your opponent can't stop you from using the ability. When you announce an ability, you announce it, pay all mana and tapping costs, and the ability goes on the stack. Your opponent can't interrupt the process of playing the ability. The first chance your opponent has to use any spells or abilities would be when the ability is already on the stack.
The reverse is also true. If you haven't tried to use the ability, you pass priority to go on to the next phase and your opponent tries to tap one of your lands, tap the transmuter, or kill the transmuter, you can use the ability in response, and thus still get the effect.
However, several people have said that the best strategy is to wait for the Transmuter to use its ability so that it's tapped, and then use a spell or ability to kill it. This is a fine strategy. I just wanted to add that by this time, the ability is already on the stack, and even if the transmuter is killed at that point, that activation of the ability will still resolve normally and you'll get to put an artifact from your hand into play.