Suggestions for a circuit/off the shelf component/device that can amplify an ac signal from 12V @ 5ma to 300V @ .25 A

This feels like an XYproblem in the making. What end task are you trying to accomplish. 

Its for signal conditioning for a relay (utility power application).  The output of a potential transformer (12V input) goes into the amplifier which outputs 300V.  This is fed into a SEL: 351S relay.

What's the actual goal you're trying to achieve (i.e. if you could start at the beginning, before getting to your immediate objective).

Just trying to guess, it sounds like some sort of industrial power system testbed to simulate faults or something, that you're trying to achieve?

I doubt you're going to find anything off-the-shelf unless it's a well-known problem you're trying to solve perhaps. However I'm sure you could get an engineer to put together something custom, if you were able to explain it a lot better.

What's the actual goal you're trying to achieve (i.e. if you could start at the beginning, before getting to your immediate objective).

Just trying to guess, it sounds like some sort of industrial power system testbed to simulate faults or something, that you're trying to achieve?

I doubt you're going to find anything off-the-shelf unless it's a well-known problem you're trying to solve perhaps. However I'm sure you could get an engineer to put together something custom, if you were able to explain it a lot better.

I updated the original post to try to clarify things.

That's a lot more feasible now. But still I don't know of anything off-the-shelf. Probably if you were constructing it from scratch, your solution will require a transformer, either to step-up, and/or to isolate all the outputs, since you may not want a common-ground for your tests for all outputs (I've no idea).

I don't know if there's a better method, but if it were me, since you're going to need some sort of amplifier or buffer anyway, I'd investigate building an audio amplifier output stage with complementary transistors, since they are available to support several hundred volts, and drive it's input from the signal source. Even if it didn't reach your 424V that's not an issue, since half that would be sufficient to drive a 1:1 transformer in push-pull using the amplifier stage on each end of the primary winding. A 1:1 or 1:2 transformer should be possible to find. You may get distortion of a few percent though. You may also need to protect the input side depending on what you're driving it with.

Your transistors would likely dissipate a Watt or two, so multiplied by all your channels, such a solution would require a large heatsink or fan, and would therefore be like a 1- or 2-rack unit sized piece of equipment. 

In summary, I don't think it's too complicated if some distortion is fine, and wouldn't be too expensive in terms of material costs either (maybe $1k materials since you'd be purchasing in low volume), but you'd still need to pay someone to design it and construct it (not me). If you're interested to self-design it, you should look at audio amplifier circuits, but you might not find a ready-to-use design to match your specifications (hence the need to design).

The above assumes you wanted ten outputs. But you've edited your original post, so now I can't check if that really was the case, or if I had originally misread. It's good etiquette not to delete content, just to append with (say) text indicating the additional content. Half the comments from people make no sense now that the original post has been edited.

Sorry,  yes it was 10.  I was trying to simplify things, but I get your point.