I know your concern isn't just when using EAC, but as an example, I use EAC somewhat frequently (especially to convert my audiobook CDs to MP3 files), and have left avast! running without any problems (and when converting an audiobook, I may process 10-20 discs in a row). I probably wouldn't run a scan while I'm creating the MP3 files, but that's just because the scan can use enough resources (system & disk) to slow down EAC. If you're still concerned about avast! interfering with the extraction, I'd choose the temporary disable right-click as mentioned above (just remember to turn it back on, or disable it for an hour, and then "re-disable" it again if need be).
FYI, I'm running Windows XP Pro with 2GB of RAM, and besides avast!, I normally have at least one browser running (often two) at the same time I'm running EAC.
By the way, slightly off-topic: You mentioned that you will sometimes turn off your internet connection. Have you used EAC with an active connection? Polling the online database to see if your disc is already cataloged can be very helpful - it keeps you from having to manually label your tracks (assuming you choose a reliable entry, sometimes you may have to try a few of them), and this can save a lot of time. It also compares your converted files with the online database to see if the extraction was completely error-free. Finally, after extracting you can upload your own data to help other people extracting the same title later on (but that's up to you).