The unofficial rulebook states this for prone; "Cannot play hero cards or spend MP. Heroes stand up during the Status Results Step but cannot play actions or reactions or spend MPs during the next Hero Cycle. They may still play interrupts during the Darkness Cycle"
Based on that, yes, they can dodge, since it is not an action, reaction, or movement point.... or is it?
I haven't run into that yet, but I'd probably house rule that you can't dodge when you are laying prone on the floor.... unless you are in the middle of performing 'the worm,' then exceptions can be made.
In order to make time for writing I must give up working or gaming...
I don't let monsters use any of their Fate Recipes while Prone - that seems broadly equivalent to Heroes who can't play their Hero Cards; it also usually produces common sense results (why would a Prone monster be able to dodge anyway?)
Moreover, it's not easy to knock a Monster Prone - you need to fulfil a Fate Recipe or play a special (and rare) Hero Card. It seems to be appropriate that knocking a Monster Prone be a "set up" for a further action - knock them down, then hit them. As such, depriving a Monster of a key defense - like a Cyclops' dodge ability or an Undead resurrect - seems like the main reason why you'd go to the trouble of knocking them Prone in the first place (merely getting them "out of your hair" for a Cycle doesn't seem like much of a recompense).
Edit: BTW, it's only Fate Recipes I disallow; automatic powers still work. So a Prone Yardu still needs two successful hits to injure him, but can't summon Soulless.