You could do it w/o a transformer. It would be easier and much more versatile than the Class E designs.
so if one were to build a class C AM transmitter using a solid state BJT or FET or more plus a transformer for modulating the B+ at a 50w or so level could it be done ?
I'm not going to agree with the anti-iron naysayers.
It could be done. The only real obstacle is finding a usable transformer and deciding on the audio output stage type to be used.
Either push pull requiring a center tap on one of the windings or totem pole and capacitor coupling the stage to the transformer.
It's fairly easy to see that if 12V is being used, a 24V CT transformer would be a decent place to start for a push pull stage's winding. For the 'secondary', a slightly higher voltage would be in order just like the tube stuff. I've seen alot of xfmrs floating around with 18-30VCT and 30-45VCT widnings, each rated 4-8A. What to do with the 120V winding? feedback maybe. or ignore it. Feedback or some kind of frequency EQ should be considered when using power iron for audio but that is not hard to do. This does noit have to be hi-fi, just clean over the audio range. The low end is not an issue since the iron is 60Hz. The high end of 3-5KC might need some attention and some loss in a power transformer can be expected up there. If you find one rated 50-400Hz, that would be better.
Another source might be the output transformer from a 50-100W transistor PA power amp. There are some of those floating around in broken condition for a few bucks. In fact, repairing one and transplanting the guts to your project might be worthwhile. Those are often tapped autransformers which would be perfect for selecting your choice of modulation voltages.
The other challenge is the question of DC magnetization of the winding used as the modulating secondary because power transformers are not designed to carry DC in a winding.
A good thing to help here would be to use a xfmr rated 2-3X the necessary power level. big core won't saturate as easily and a 150VA transformer is still not very large.
The same technique as used in tube stuff could be done, finding a suitable choke for the DC current of the final and cap-coupling the mod winding to it. 5000uF is about 0.3 Ohms at 100Hz so the cap is easy to find, even a 20,000uF for some overkill. Chokes can be had from old low voltage high current power supplies such as motorola base station units. It might take a coule in series. It's not unusual for those nasty old ferroresonant DC supplies to be choke-input. The choke is often hidden under the chassis.
So, what voltage for the B+? 12V? 50W input - would be about 3 ohms. 10mH is 7 Ohms at 100Hz, and I bet there are some 20-100mH 4A chokes or better out there in junque if looked for. I just stripped out some 24V supplies from RCA film chain projectors to get the rackmount chassis&panels and the chokes were 8A. I have not measured the L yet.
OK well there is a budget for space requirement for the iron so that's what it comes to if that is a concern. Hopefully, you will succeed in building a "heavy metal" solid state rig.