The trick here is to recast the problem as a series of smaller problems that we know the answer to. Sure, we could apply Ampere's law directly, but that means we have to do two integrals (one for the current and one for the B-field) and those integrals may not be trivial. Instead, we'll reduce it to solving a single integral. I'll give a general description in text but also include an image that has a diagram and equations.
Let's start by imagining that this spinning disk of charge is actually a stationary disk with a bunch of current carrying wires on it. Essentially, we'll cut the disk into differentially thin hoops and make the connection that whether current is flowing or charge is spinning makes no difference to us for this problem. The amount of charge on our hoop is equal to the charge density times the differential area of our hoop, 2πσrdr. We are given the angular speed of the disk, and from that we can get the period of rotation. The charge divided by the period will then give us the "current" in our little hoop.
The on-axis value of the magnetic field due to a loop is a relatively simple solution that's usually given to us in the text. If we pop in our values for current, now we have an expression for the field due to our little loop. Make sure to note that the radius in our expression is little r, radius of the hoop, not big R, the radius of the whole disk.
Now, to get the field of the whole disk, we just have to integrate the radius from 0 to big R. The integral is of a form that shows up all the time in E&M, so you probably have it listed as an appendix in your textbook. If not, it's relatively simple to integrate via a u substitution.
https://imgur.com/a/YIXCN0z
