I am a guitarist, so I'll often write about guitars and basses. The information in this article generally applies to any kind of pickup, whether it's in a B3, a clavinet or an electric piano.
Pickups are an entire topic on their own. There are several forms of pickups that can be made, but by far the most common kinds are high-impedance, inductive sources. Some guitars include on-board amplifiers (called active pickups) but the pickups themselves are still usually passive.
There are also piezo pickups, which use a piezo-electric crystal to measure the vibration of the body of the guitar. These pickups can vary greatly in quality, with many of the low-end versions providing a tinny, uneven response that is far inferior to a microphone. These crystals are best modelled as a very small capacitive load, meaning they are effectively high-impedance (at least at audio frequencies) and disturbed by small amounts of current loading. Most guitars with piezo pickups include built-in 9v amplifiers (sometimes called buffers, I usually reserve the term buffer for a unity-gain amplifier in electronics), often with some kind of signal conditioning. Newer machines even have built-in DSP for effects processing.
Of particular interest to me are low-impedance coil pickups. They are simpler than standard pickups and provide a flat frequency response, which is useful for DSP, chimy-clean tones and bass. A beautiful, though expensive, example is the Alumitone, though they can be made fairly easily with any material of choice. For example, it would be easy to DIY by cutting some copper foil into a pattern, or using a loop or two of wire around an iron bobbin. You could then solder to the foil or crimp to the wire. Then you'd encase the pickup in epoxy for durability. Now that 3d printing has become super common, it would be easy to prototype a pickup with plastic and foil, or even using conductive filament. Chinese manufacturers also provide inexpensive CNC services, which would be useful for the final prototypes or for production.
Modelling inductive pickups is tricky. The magnetics of the coils interact with the strings, eddy currents are generated in the strings, the core and the magnet wire. There's a parasitic capacitance between each wrap of wire. All of these factors play a part to influence the final sound of the pickup. With that said, there are some tricks to characterising pickups empirically as well as getting a "good enough" value for SPICE simulations. Low Z pickups are less of a concern, they're designed to be audibly flat anyway.
First, empirical measurements can be made with a Vector Network Analyser, or VNA. VNA are usually meant for antenna characterization, but there are instruments such as the OMICRON Bode 100 that can do 1-40 MHz measurements. It's an expensive bit of kit though, and we can probably get good enough measurements with far simpler instruments. The basics of building one of these machines can be found in DIY VNA with Pmod XLR