Based on interviews, Lucas Arts were using common MIDI tools and composition software.

The real magic happens in their music driver, which takes the sound effects, multi-track MIDI music, along with track priority infomation and dynamically down-mixes it to however many tracks the current audio device has. Just one in the case of the PC speaker.

I assume they would have had a setup that allowed them to quickly hear what the track sounded like on all their target audio devices.

I don't know of any toolkits used by the pc speaker games. I did enjoy some videos on YouTube from the 8-bit Guy recently that had some good info though:

https://www.youtube.com/watch?v=q_3d1x2VPxk

By the late 1980's early versions of Cubase et al. existed. These were not yet DAWs in the modern sense(while there were some very early high-end examples of multitrack digital audio, that phenomenon waited until memory and storage were cheaper) but they were competent sequencers and could drive MIDI devices easily. Building up a MIDI sequence and then exporting that to the game engine format was the preferred workflow for the studios on PC attempting full scores(rather than "tunes and jingles") starting in this timeframe since it offered the flexibility of hiring traditional composers who could record in from a keyboard and then do some cleanup and edits for the target device as needed, including simplified beeper versions.

The tracker music/custom sequencer formats operated in a parallel universe alongside the MIDI workflow and were more often the provenance of scrappy demosceners and independents who saw an opportunity to completely control the output quality(as long as it was sample-based). Not everyone literally used a tracker type of workflow and there are examples like MML(Music Macro Language) as another idea of source formats, as well as the low-level "enter hexcodes in a machine language monitor while the playback routine is running" (used by some C64 composers.) If you played a DOS game made with QBasic it probably used the PLAY statement to control the beeper, with an MML-style syntax. This style of syntax would appear again with programmable mobile ringtones.

In the mid-90's the balance shifted again towards CD audio, ushering in simple drum loop sequences as the quick-and-dirty audio filler of choice, and everything since then has largely been variations on that theme with more tracks and processing.

I read, back in the days, "trackers" were used for electronic music composition. But I don't know if that was the case for game music.