The short answer is NO. If you care for an explanation then read on.
To get AF lock on the subject when in Live View mode you have to use the AF-ON button. I just checked the custom functions on my 50D and there is no option to map the AF to the shutter button when in Live View.
I'm not sure why you would want to use the shutter button instead of the AF-ON button. Is it because of the convenient placement of the shutter button? I personally find the AF-ON button conveniently placed right under my right thumb. Or is it because you want to use one single button for AF and shutter release?
Live View is designed for still subjects which do not move, not for live action photography. You may find the AF to be very slow in Live View. This is because it uses contrast based focusing (Live mode). If this is the reason why you want to use the shutter button instead - to get faster focusing - then you can try changing the AF mode in Live View to Quick. This way the camera will focus using the AF sensors. That's the same kind of sensors used when you auto focus when shooting through the viewfinder with the shutter button.
The Quick mode gives you much faster auto focus in Live View, compared to the Live (contrast based) focusing. Although you will still have to press the shutter button to take the picture.
The story of the AF-ON button began in early 90's or late 80's. Back then it was known as the "back button AF". Most SLR cameras in those days were still analog and didn't even have the Live View because they didn't have the LCD on the back. Canon was the first manufacturer to introduce this. They allowed you to move the AF function from the shutter button to the AE lock button (asterisk) so that you could auto focus with one button and take the picture with the other. Then they introduced the dedicated AF-ON button for this.
You can still configure your 50D to operate this way, even when shooting through the viewfinder. You will find these settings by going to Menu, C.Fn IV, first page.
This is the way Live View shooting works by default, and there's no way of changing it. And as you may understand now, this is something that people actually want, most professional photographers are using not one but two buttons to take pictures even when shooting through the viewfinder. They use AF-ON for focusing and the shutter button for shutter release. This has several advantages over conventional shooting with just one button.
One advantage is that it's easier to lock focus. Say you want to take several portraits and you want them off-center. The AF-ON button makes it easy. Focus on your subject by pressing the AF-ON button. Then take your thumb off the button and re-compose your shot to move your subject off-center. With AF removed from the shutter button you can take the shot any time you like. And you can remove your index finger from the shutter button after a shot and the person is still in focus. Pressing the shutter button again doesn't affect the focus.