Actually from what I've read quite the opposite - DO NOT get a slim/low profile filter for the 12-40. A few people have reported the filter glass touching the center of the front element and leaving scratches. Seems to vary a bit from either lens to lens or filter to filter. Search some DPR threads and you can probably find details as to which manufacturers in particular have caused problems like that.
As to vignetting I haven't tested myself, but looking at the lens, hood design and entrance pupil I'd say you should be just fine with a standard filter. You'd probably have to stack at least two filters to start to have an issue.
Using a B+W KSM CPL in XSDP mount (slim) on the 12-40 and loving it - no vignetting. However will now check the front element!
OT - It's a shame Xuma don't make 62mm magnetic adapters though - when changing between inside and outside, bright and dark conditions, it's a pain to keep installing and removing the filter. It would also add a little bit more distance to the front element.