Sure you can. Get, for example, a PR2 robot and program it accordingly. You can also use simpler robots, but in all cases, you will either end up with a ridiculous amount of cost, or with a very unreliable performance. Not mentioning the hours of programming time you will have to spend on the project.
You may be better off buying a coffee maker that you can control remotely, e.g., via Wifi. Use the app to start the coffee making process once you are awake, or program your phone to detect that and start it for you. In industrial settings, robots are almost never the answer for tasks you can do without (e.g., button pressing). This is not so much different.
(I am not affiliated with any of the producers of the products linked to in this post.)