The answer would be "Of course!" and "Of course not!".
You don't need electricity to run a "computer". But as soon as you're trying to hook up this computer to the Internet, serving responses on an electrical network... I can't imagine how to do this without electricity.
Of course you're not looking for answers using a turbine, are you?
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2917228/
But: it takes a full 10cm wafer to make 50 D flip-flops(a basic component equivalent to a memory bit), and it works at frequency of 0.5hz .
So it's a challenge.
But maybe it's possible to build a very dumb terminal with this(say keyboard + screen
https://www.nature.com/articles/s41378-018-0018-1
) , if you offload everything to the cloud - and use a really basic communications protocol.
But will the Amish accept a computer/network ? What will it look like ?
Mechanical logic is an interesting option as some newly developed MEMS logic gates might be able to operate at MHz frequencies[1].
[0]https://apps.dtic.mil/dtic/tr/fulltext/u2/a084924.pdf [1]https://www.nature.com/articles/s41467-019-08678-0
But bitrate would be low.
Some things seems like they could become very easy - e.g. addition is just pouring the water from one location into another. Overflow becomes literal overflow...
Subtraction becomes a problem of opening a valve until you've removed the right amount of water... Maybe you could use a weight mechanism that'll close once the removed amount equals the amount you're subtracting?
A precise one and/or a fast one certainly would be very hard to make, and I'm sure the real ones were a lot more complicated, but many simple operations seems like they'd be quite easy to figure out.
Can't imagine it'd be remotely practical at any kind of scale, though... It's amazing they were used as much as they were.