I am studying figured bass. I see some chords might be incomplete, for example 8/3 chords. Would those chords get Roman numerals assigned to them, or must all 3 notes of a triad be present to be able to assign a Roman numeral to a chord?
No, not all three notes of a triad must be present for it to be given a Roman numeral.
Common practice requires at least two pitches to be present to recognize it as a triad: the root, so that we know what the root of the chord actually is (!), and the third, so that we know the quality (typically major or minor). This means that the chordal fifth is typically the chord tone that is omitted.
The same logic, by the way, applies to seventh chords: the root, third, and seventh are necessary, but the fifth can be omitted.
Standard voice-leading procedures, however, suggest that incomplete chords should only occur in root position. If a chord is inverted, all chord tones should be present.
Figured bass and Roman numeral analysis are two separate things.
In figured bass, each interval above the given bass is specified. A chord that features an octave and a third above the bass would be figured 8/3. This is especially important if it's the composer's specific intention to have an incomplete chord. Figured bass does not employ Roman numerals.
In harmonic analysis, each harmony (as opposed to each chord) receives a Roman numeral. As long as a set of notes comprises a new harmony, regardless the number of pitches present, then a Roman numeral is assigned. Harmonic analysis does employ figures in order to clarify inversions and ornamentations.