They are always available. On the magic you can get veggie burgers at Pluto's on deck 9. I haven't been able to get them through room service unfortunately. Additionally I have never tried to get them in the main dinning rooms (as there is so much great veggie food). At Castaway - you can ask a server to get one for you as it is a buffet and they keep them behind.
As far as the separate griddle - I haven't asked as it doesn't bother me. Disney is really great with allergies so imagine they would do it separately for those with allergies/religious reasons. Hopefully someone else will have the answer for you