Can an enemy Disposal Orders my trash before I choose to play something from my trash with Fizz?

They can't force you to choose first and then strike in a "blind" window — but the practical answer is yes, they can still stop Fizz. Here's the timing:

When Fizz's "When you play me" trigger goes on the Chain, that's the exact moment you pick the spell in your trash (FAQ #10020). There's no gap where your choice is pending — you declare the target as you add the trigger, and that choice locks it in.

What your opponent can do:

• After you've declared your target (the normal play): they respond to Fizz's trigger with Disposal Order, recycle the targeted spell, and Fizz's ability fizzles — with no valid target, the "play that spell" instruction is ignored (Rule 359.3.e.7; FAQ #10771). The spell never gets played.
• While Fizz is still pending on the Chain (before his trigger even exists): Disposal Order is a Reaction, so it's legal, but they're guessing — you haven't chosen yet, so they'd be wasting it if you pick a different card.

So they can't snipe your choice before it exists, but once you name the card, Disposal Order in response makes Fizz do nothing. Your only real protection is having the spell be un-recyclable or answering their Disposal Order first.