What vowel sound does herd have?

It depends somewhat on the dialect.

In US English, the E is pronounced as a short U sound, as in (hurd). This is an R-influenced sound but not very different from other short U (uh) sounds.

In British English, the sound is (hehd) because the R is not pronounced.

In some dialects, notably Australian, the sound may be classified as a "schwa" sound, because it is midway between an (eh) and an (uh). However, the schwa designation is normally used only for unstressed syllables.

In at least one R-less dialect, the Word sounds much like head.