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It seems possible, if you apply enough knowledge, time and resources to the problem.
The cheapest way would be to do it yourself using the cheapest possible parts.
Get the requisite tools, like a WiFi shield (goto sparkFun)and a conventional arduino like UNO and goto xively.com(in case you want to control your gadget from anywhere in the world) or if you just need to control it using WiFi within your establishment only then you can follow the webserver samples code given with many wiFi shields.
Do you have the capability to fabricate the mechanical parts ?
The arduino wifi library comes with some sample code. One of them is a server code. You can start from there. Get some LEDs in place of actual motors. Turn on/off some LEDs with arduino wifi serving a web page to your PC web browser while you learn.
There is a web technology called AJAX. You may want to look into that one. Otherwise, every time you issue a move command on the web interface, you have to press "SUBMIT", kind of annoying.
I've personally tried some AJAX with arduino. Nothing too difficult once you get through the few basic sample codes and read about AJAX. I don't have a web programming experience BTW.
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The Wifi Shield is pretty expensive compared to the Arduino Yun (which has Wifi-Built-in). Is it also possible with this one?
Not necessarily ,you can get a RN-XV wifly module from Roven networks and it should be around 30 to 40 dollars max for single retail qty amounts and moreover it has UART comm. to and fro to configure the module iteself and also communicate.
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So I could only buy this and make everything without any additional hardware?
WRL-10822
well an arduino and a xbee to standard header breakout board so that you can get your XBEE over breadboard and use in the circuit OR get a xbee shield for arduino and an RNXV module offcourse!
this will give your arduino wings! then whatever you want to control or actuate you have to get that in addtion to all tihs.
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Why doesn't the Arduino Yun work for this?
I never said yun didnt worked , it works but if you get your math correct then I think a breadboard arduino and RN_XV will cost around 40dollards wherean an YUN is like 72 dollars.
I personally would never buy a yun , will rather get a pcDuino or other similar linux SBC with WiFi.
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No, here is everything more expensive than the Yun. But will it work with 3.3 V?
I think its very easy to add up as I told, I'm do not want or have time to argue for solidifying facts.
Yun is 5v compliant . It can sport any 5v shields.
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The power supply from the Automower is 3.3 V. But the Yun needs exactly 5V or will it accept 3.3 V like the Uno?
Its operating voltage for the ATmega32u4 is 5v so you need to use a Logic level converter like the ones available with sparkFun.
for more of Arduino Yun ,you can see on the Arduino website and this link >> Arduino YUN (YÚN) : ID 1498 : Adafruit Industries, Unique & fun DIY electronics and kits
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Never heard of a 3.3v mower (lawn mower?), but the below is a simple setup for a wifi bot.
Is there some sample code i can use for this?
I think yes the sample code would be there within the arduino ide that supports programming the yun, I think the arduino ide 1.5.X ownwards sport it rest just visit the Arduino Yun official page there it is defined.
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Yes the rnxv modules from roving network have gpios which can be controlled over uart and also directly from the script , read their whole datasheet , they have a real good deeply defined datasheet.