Risk - Play by mail/email
Unfortunately, play-by-mail isn't really an appropriate medium to play Risk, for a number of reasons.


Gameplay is too long, even when just considering playing in person.
while only one player can "act" during a turn, defending players get to roll defending die on each attack
choices to invade where, and in what quantity, are made by evaluating the whole board--something that can't be done if played simultaneously.


Gameplay is too long

Risk can, depending on the players skill levels, take several hours.  This is with immediate feedback from rolls and direct communication. To turn it into mail adds months to complete a single game.  I'm not sure a single game of Risk can maintain engagement over that long a time.  Even with email being much faster, it still suffers the other below issues.

Defending players

It is definitely possible that the defending players' rolls can be taken by the player staging his attacks, but this has a number of drawbacks.

The defending player doesn't get the enjoyment of defending his own territories with his own rolls, and thus all he/she gets in return is a summation "sorry, you lost the following territories; they are now occupied with my armies in these quantities".

It is a necessity that the attacking armies do the rolls for both the attacking die and the defending die (whether manually or assisted with a phone app or website). If the defending player is given the opportunity to make his own rolls, that actually means EACH attack requires a round-trip of correspondence: one to declare the attack, how large, and what the rolls were; the other to defend the attack and what the rolls were.

Only after it has resolved could the attacking army know whether to press the attack, to give up, etc.

Simultaneous play

Even two players couldn't decide to play simultaneously, much less more than two.  Take for example, two armies incidentally and simultaneously attacking a third party's territory.

Whose resolves first?  This is natural to playing on a board, where you might just go clockwise, but simultaneous play will have to 'emulate' what it means to go clockwise then--and if so, then it's no longer simultaneous: whoever falls lower on the priority of his turn--his actions are pretty much null and void.

Consider players A and B attacking player C with only 1 defending unit.  If A and B both send 15 armies after C's 1 unit, and A is determined to resolve first, B's entire plan of taking a territory, then adjacent territories no longer is applicable.

Conclusion

In short, the nature of Risk is synchronous and to modify it to play via a medium--mail or email--is too much work for the reward.  If none of these are compelling enough reasons, even just consider the setting up the board: do you really want to send an email after each ONE army is placed, and have dozens or hundreds of emails (with their delay) to get the board set up?