In my opinion, it depends entirely upon the intended audience.
For a business email, especially one in a 'formal' office (e.g. a government office, a Fortune 100 company, or any office that has people regularly appearing in suit jackets) it is, I think, entirely appropriate to use "whom."
For a personal email, the expectations are rather lower; it depends on your audience and your relationship with them--regardless of my ability, my SO and I have built a sort of private jargon that bears little resemblence to properly written English; "whom" does not often appear in our private correspondance.
For a business email in a less-than-formal setting (or an informal email between close business associates in a more formal setting) similar guidelines may apply.
As I work in an office more to the formal side of the continuum, I have been known to use 'whom' on occasion; nobody has complained yet.