Did any computer use a 7-bit byte?

The well-known IBM 1401 technically had a 7-bit byte (plus parity). It was designed around the common format of IBM punched cards, which it was designed to process; these had ten "digit" rows and two "zone" rows, of which one digit and optionally one zone (for which the zero row also counted as a third zone) could be punched simultaneously in each column.

This essentially meant that each column of a card contained a value that could be encoded in 6 bits - 4 representing a BCD digit, and two more indicating the zone (no zone, first zone, second zone, or zero row). To this, the 1401 added a "word mark" bit for internal use, for a total of 7 bits.

The second-generation Soviet computer Minsk-32 (the series size is 2889 machines, 1968-75, civilian use, one of the rare early mainframes noted for use in Antarctica) used a 37-bit word and 7-bit representation of alphanumeric characters (5 in a word). Yes, the concept of "bytes" is difficult to apply to a similar old computer (which continued the line of vacuum tube machines), but special commands for the convenience of operations with 7-bit blocks took place in the command architecture.

The ADAU1701 is a 28-/56-bit DSP for audio processing. CHAR_BIT is probably 28 on that platform like most odd-sized DSPs but I'm not quite sure since I couldn't find its programming manual

Yes; there have been several (although, to my knowledge, none in the most simple sense where seven binary bits are treated strictly as as a base-7 system of Peano-like numbers). Instead, they are systems in which at least one (typically, two or three) carry are treated as separate state-modification bits.

The most oldest/most simple example (although it may not meet the definition of a Turing-complete computer) is the ancient 5/2 abacus.

More recent examples generally are cases where some form of binary-coded decimal is used, particularly those that use Chen-Ho encoding (which fit a better conception of the system being "7-bit", as Boolean logic/operations can still be (relatively) easily applied, as opposed to more packed (or packed/padded) 7-bit numbers, which require a variable number of instructions to ascertain certain binary/two's complement values.

Of these, the "two of seven" approach is most common. Examples: the IBM 650, the FACOM 128, and the "IBM 370 compatibility feature" (hardware emulation) built in to the IBM 7070/7074.

The VT52 text terminal certainly doesn't qualify as a full computer, but it does have a processor running software out of a ROM. The RAM holding the displayed text is 2048 7-bit bytes. The character generator ROM is also 7 bits wide.

The Norsk Data ND-505 had a 28-bit address bus.