Speaking professionally:
I'd start with a good, rich celery juice. Get as many celery leaves as you can (they have much more flavor) and stalks together. Purée them in a blender, letting it go for a while to get really broken down, then strain it, pressing as much liquid through the sieve as possible. Do not attempt to reduce it, or cook it in any way - that's going to ruin "the taste of a flavorful stalk of celery". If the raw juice isn't flavorful enough for you there's not much else to be done, unless you have a rotary evaporator handy.
Now, here's where things go a bit astray from what you're used to…ss you've already learned, you can't make a cooked sauce with extra additives; even just using a roux or cornstarch would require heat, which we've established as a no-no.
So - we're going to need a bevy of "modern" thickeners. I'd start with a mixture of xanthan and guar gums; while they both work well enough on their own, combined in a 1:1 ratio they're significantly more effective. You can add up to 0.5% of the weight of your celery juice, but that can get a little snotty so I'd start at 0.3% and work up. It won't be catsup thick yet, but it will be viscous and resistant to flowing suddenly - good backbone aspects of a squeeze sauce. To give it more body, bring it closer to your ideal, we'll add some modified starch. It does the same thing as cornstarch or flour, but it's been altered so that it doesn't require heat - you just put your juice in a blender and sprinkle it in until you're at the desired thickness. Since we're looking for thickness above all, I'd use Ultra-Tex 8. I can't give a percentage for that, as your desired outcome is up to you, but as it won't get ropey or gooey if you use too much you can just wing it.
Those two alone should suffice to at least get the juice thick enough to use as a condiment, but it would be very, very smooth - like a cream sauce as opposed to a pureed vegetable product. If you want that extra bit of texture, as a final touch we can blend in some psyllium powder, basically very finely ground Metamucil. Not too much, I don't have a percentage handy but as too much makes it gelatinous I'd start at maybe 0.5% and go from there. It will add that well-cooked-vegetable-matter texture that catsup has.
To finish it, add celery powder (not celery salt or seed) to taste, and any other seasonings you may wish such as black pepper, garlic powder, onion powder, MSG, red pepper flakes...pretty much anything but salt.
ETA - all ingredients are available on Amazon, or Modernist Pantry if you're averse to Mr. Bezos.
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