"Background music apart from the songs".. That wouldn't be called SFX then, it would be called BGM. Sound effects are walking, doors, ambience, etc. So your thread title is incorrect.
Answering your question:
Legally or illegally.
Usually, the OST will be released with the Blu-Rays/DVDs.
I looked around a bit and found this comment:
If you don't want to wait until the end of October for the real BGM, you can just use substitute BGM. Try to find something that sounds similar or works just as well as the original (not too hard). Looking for music by the same composer might help. You can also look for music that has a similar genre to AKB0048. I haven't watched that anime, so I can't give any suggestions. Be prepared for people saying "Is that music from (insert other anime here)." I've gotten that so many times..
By the way, you can't request for music downloads on the VAA as explained
here.
Now, if you do need help with SFX, download the many files users of the VAA have made, go to SFX websites, or rip SFX from the anime itself. Rip only sounds that don't have talking/BGM in it.