Is there a way to have the Mac App Store check for updates automatically?
The App Store doesn't offer that functionality by default, and you can't configure or script it to check for updates. So...

The closest you're going to get is automatically opening the App Store so you can look at the icon to check for updates. Here's how to make the App Store open automatically at an interval:

Open iCal and add a new event. Call it something like Check App Store for Updates. Double-click it on the calendar and click the Edit button. Set the repeat menu to Every day or Every week.
You can get it to check at some interval in between (say, every 3 days), by using the Custom... option.

Set the alarm menu to Open File. In the newly-showing second menu under alarm, choose Other... and select the App Store in your Applications folder. Set the last menu under alarm to on date.

This way, iCal will automatically open the App Store at the interval you specified under the repeat menu. Its window will pop up to remind you to look at its icon. When you do, you'll see the number of available updates.