I know that due to errata, strategic planning only lets you ready a support without mods.

However, does dogfight which exhausts a support and readies a support have the same treatment?

do you have to choose your own support that has no mods?

Dogfight has no erratum. You should note however that it says vehicle not support and if you do not do the first part then you cannot do the second part so unless your opponent is playing vehicles....