I understand that an RF modulator was used but what kind of circuitry did it use to feed the RF modulator?
To start with, the TRS-80 did not use an RF modulator. It delivered composite video (*1) as it was meant to be used with its own brand monitor - which basically was an RCA TV set with the RF part removed.
The TRS-80 Color Computer used an MC6847 IC but I can't find any references to what the original TRS-80 used.
Like many others, just a bunch of counters and muxes (*2).
Was it an IC or did it generate video in a similar way the Apple I or Apple II did?
Somewhat, except it was rather straightforward to make it simple instead of squeezing out the max as with the Apple II.
A schematic would be very helpful as well.
There's even something way better, the original TRS-80 Technical Manual which includes a detailed description of each stage (*3).
*1 - Well, not really, as the output signal wasn't a standard (1V) composite signal, but generated and transferred at TTL level (5V). The TV set had not only the RF part carved out, but also a little board with an optocoupler added, which not only transposed the signal in level (Again not standard, but this time the 2V signal the remaining logic wanted), but also served as shock protection.
*2 - After all, a 6845 or similar is nothing other than a bunch of counters integrated into a single ... well, Integrated Circuit :))
*3 - But there are, in addition, many sources for TRS-80 schematics online available ... like this one.