"Musical Theater" is as broad of a category as they come. From the almost primitively classical works of Gilbert and Sullivan to modern avant garde styles that defy analysis, and every classical and popular style in between. You'll need to specify a composer or show that you want to sound similar to in order to get a specific answer.
One thing I'll point to that's very common in modern pop styles, and isn't so common outside of MT is the use of a IV chord over the 5th scale degree, which resolves to I. So in C major, that would be an F chord with a G in the bass (resolving to C). You could analyze that in a few different ways:
- An add9 chord with the 9th in the bass
- A plagal cadence in the treble with an authentic cadence in the bass
- A heavily extended/modified/suspended V chord
Whatever the analysis, the result is a chord with a medium amount of tension that resolves with an amount of pull somewhere between the authentic and plagal cadences.
Edit: I just saw your edit.
So yeah, the chord above is a big part of those genres. You also see a lot of highly extended chords, in particular 6ths (call it a 13th if you like), 9 or add9, or both (6/9). Lots of first inversion chords to make stepwise bass patterns (e.g. I → V6 → vi(7)). You also see lots of suspended chords, often not resolving as they would classically but instead just substituting for their "normal" versions. In general, crunch things up a bit by sticking some non-chord tones in there that are only a bit dissonant.
Another huge part of it is rhythm. There's lots of syncopation in the accompaniment. A few common patterns:
- Accents on the upbeats of 3 and 4, often with the chord change happening on that upbeat of 4 (that is, anticipating by a half beat)
I'm away from my computer with notation software installed, and I don't want to go digging through what I can find on Google, but if you're interested I can jot down some brief sketches illustrating all of the above tonight.