I know that "modulo" is a preposition. However, I'd like to say something equivalent to, "We use each multiple of ten to modulo that digit of the number." In other words, I'd like to say "to take the remainder of by using the modulo operator" in a way that's grammatically correct but more terse, if possible.
No, modulo is not commonly used as a verb.
However, in mathematical usage, the verb used to describe this action is to reduce modulo x (source).
There are a few ways to phrase this. The verb can be used transitively, where someone is performing the act of reduction on a number. Or it can be used intransitively, where the number itself it reducing.
We reduce 13 modulo 10 to get 3.
13 reduces to 3 modulo 10.
To re-work your sentence, I would write
We reduce the number modulo each power of ten to get the respective digits.
This phrasing is quite idiomatic, at least for mathematicians. It both uses modulo as a preposition and is fairly concise. However, it might be hard to understood for non-mathematical audiences. To avoid technical jargon, I might write
We divide the number by each power of ten, and use the remainders to get the respective digits.