Microwave Oven: Switching Magnetron (Transformer) with SSR
Normally the transformer feeding the magnetron is just switched on/off by a relay or triac. A relay or SSR capable of handling a somewhat inductive load would be easiest. 

Here's a schematic of a very simple old-school microwave oven. Yours may have such a schematic stuck internally. Possibly if you added the schematic and some photos someone might be able to give you a more precise answer. 



If you only interrupt the circuit going through the timer switch and don't muck with the interlocks and fuses it might be relatively safe to do this (from the pov of safety from leaking microwaves), but keep in mind the lethal voltages internally (both the magnetron high voltage and the mains), and the potential for fires.  

There are many opportunities for serious injury or death in this endeavor, so I have to suggest that buying a new $50-$150 microwave would be a better course of action. I have modified a consumer microwave oven to create gaseous plasma inside a vessel but it takes quite a bit of knowledge and care to do this relatively safely.