Malwarebytes can wake up my computer at 4am, update itself, run a scan and then let the computer go back to sleep again.
Can Avast Free do that?
If so, how would I set it up?
I am using Windows 7
Short answer is no. avast will automatically update when you restart your machine to the latest vps update.
Or you may choose to manually update. 
Here is one way it can be done if your computer uses the sleep mode
http://www.screencast-o-matic.com/screenshots/u/Lh/1370274785640-96368.png
http://www.screencast-o-matic.com/screenshots/u/Lh/1370274888444-63132.png
http://www.screencast-o-matic.com/screenshots/u/Lh/1370274669599-37975.png
You can also select the Create a Custom Scan in the first Screenshot and go from there.
Whilst avast can wake up the computer to schedule a scan but there is nothing in the scheduled scan settings that would check for updates before running a scan.
So you would have to be a little crafty and have the system wake up, like the MBAM scan and don’t shut down after the update and scan. Whilst that is going on avast should also have been woken up by the activity, so hopefully avast would do an auto update check (if your auto update interval has been exceeded).
I would set the auto-update interval to 120 minutes, the lowest allowed in avast free, so the system wake up for the mbam scan should also trigger an avast auto update check.
Now you can set your avast scheduled scan start a little while after the estimate the duration of the mbam update and scan, probably 04:30 and set that as the scheduled scan time. As Bobs image shows you can have avast shut down the system after the scan. I don’t know if that is literal (shut down as opposed to sleep) as you may not want to do that, so just let it go into sleep mode according to your power settings.
Thanks for the quick and helpful suggestions, guys!
DavidR’s scheme would work fine except that it incurs a behaviour that I am trying to avoid - namely, having Avast do an update when I am using the computer - as this slows the machine down so I am peevishly clicking on stuff and wondering why it has slowed down! The short update interval would bug me. (I am easily bugged!)
I’d like it to only update overnight.
Is there any way I can use Windows 7 Scheduler to do a “manual” update and not have the auto update?
Part of what sets avast! apart from the rest are the streaming updates. Bypassing that feature would put your computer at risk unnecessarily.
The streaming updates only takes seconds if that long. If you want to have the regular update happen before you start using the computer then simply wake up your system
2 hrs before you start to use it.
It makes no sense to run a full scan daily.
Colour me confused, your subject and comment about mbam doing an update before running a scan had me thinking this is what you wanted to replicate.
I don’t believe you would gain anything by setting updates to manual as it is only the virus definitions which are auto and they are incremental, so shouldn’t take long. As Bob mentioned you would be missing out on the streaming updates as this relies on the definitions updates being on auto update.
The auto update should be restricted not to use high amounts of resources, so what is your CPU and RAM on this win7 system (as this really shouldn’t be happening) ?
As mentioned a resident antivirus shouldn’t need to do frequent on-demand scans.
There is no command line functionality, certainly not in the free version, whilst there is some command line functionality in the paid versions, I don’t believe one of them would do an update.
If a user sets up avast to “wake-up” to scan, that is exactly what will happen. Sometime shortly after wake-up avast will download
to the latest vps and could/would happen during the scan. If the OP wants to wake-up then update then scan there is no way to do that.
Sorry.