I really wish I had the jazz experience to give a more informed answer, but I will try to point you in the right direction.
One common example of modal jazz is Miles Davis's So What.
After the head with Bill Evan's chord vamp the solos work over these chords: 16 bars
Dm7 8 bars
Ebm7, and 8 bars
We can note...
- there are only two chords, most jazz standards have many more chords
- the chords aren't functional, standards use a lot of functional
ii-V-I or other root progression by descending 5th
- the harmonic rhythm is extremely long, standard harmonic rhythm is between 2 chords per bar and 1 chord for 2 bars
Frankly, the question that comes up in my mind is what does the bass player do to create 16 bars of walking bass on one chord?!? But you ask...
How does modal jazz use chord progressions?
I think it's important to understand there was a desire at that time - roughly the late 1950's and 60's - to break with convention.
If the conventional use of chords tells the soloist to change every bar, there is a certain predictability to what the soloist will do.
What could be done to allow the soloist more freedom and create a less predictable melody? One thing to do - the modal approach - is to more or less stop using chord changes to control the soloist. In this sense modal jazz uses chords to given the soloist more freedom by slowing down the harmonic rhythm and not using frequent chord changes.
A natural follow up question is: why use non-functional changes?
After playing 16 bars of one chord you certainly risk beating to death the set of notes for a single mode. (Conventional changes mitigate this through frequent chord changes where you hear a limited portion of the key per chord change.) In order to 'refresh' the tonal palette in modal jazz we can't change to a closely related tonality, as it will contain nearly all the same notes. We need a strong contrast. In So What changing the chord by half-step
Ebm7 gives a completely new tonal palette.
I think that is the general idea of how the chords are used.
In terms of scale/chord, So What uses the Dorian mode for both chords which is the conventional scale/chord pairing. I suggest listening to Miles Davis's Flamenco Sketches to get an idea of how you can break out of scale/chord thinking. Scale/chord sort set's up this notion of selecting a scale to realize a chord in a lead sheet. Obviously modal jazz doesn't require a lot of chord realization. Instead of thinking that way, we can think scale choice and treatment driven by an expressive goal.
Anyway, how to solo in modal jazz is a different question from how are chords used that seems worthy of a separate post.