If the finger moves too much, the variation becomes one of pitch as well as intensity. Vibrato in string playing is the most personal of considerations and is almost never discussed with others. Most orchestral players do not use vibrato, and it's distinction keeps the clarinet as the purest of qualities one hears in the orchestra. This was not always the case, and is not even completely true at present, but certainly enough to make the statement. Vibrato is used by clarinetists in the performance of solo and chamber music, and this use again varies from player to player and from piece to piece.
Some players use the lips.
As simple as that sounds, it does work, especially if you have sensitivity and are able to discern degree of lip movement under the reed, and of course if you are willing to start slowly, listening all the while for results, and judging those results. Some players use the actual support mechanism of the body, usually called the diaphragm. It is rather difficult to control the speed, but it seems the truest for not varying the pitch. Some players use the throat, which usually produces the strangest of sounds. There was a great flutist who was principal in the New York Philharmonic for many years, a wonderful musician named John Wummer, who used vibrato from the throat.