Re: “classical” vibrato, I agree with James, that is most usually taught as being done with the diaphragm, and one gets told off for doing it with the throat - that works too, but involves tension and constriction thereof - which goes against Daiv’s other advice in recent threads! Playing totally without any kind of breath vibrato is really quite hard, even if one hasn’t ingrained that horrible mainstream, mid C20th slow, wide, omnipresent vibrato - harder than getting a properly controlled vibrato!
On Jim’s main point, waving fingers up and down closely over (but without touching) one or more open tone-holes can indeed deflect the perturbations of the air column so as to produce a moderate vibrato - you can establish that by simple experiment. The down stroke pushes air towards and into the tonehole, the upstroke sucks it away, plus there can be a shading effect from the mass of the finger, just as keyed tone-holes have to be tuned according to the rise of the keys. The effect is weaker than normal finger vib and is most pronounced when you do it over the sounding tone-hole, not a lower one. I wonder if this is what Jim has seen/is referring to? This or the “tapping the side of the hole” method (which simply partially closes it, thus flattening the tone) are, of course the ONLY ways to finger vib an E (i.e. with R3).
Normal finger vibrato is done by tapping the next open tone-hole below the one that is sounding, which has the effect of flattening the note as in a cross fingering. Note that there can be no sharpening element in such a vibrato, unlike a breath-driven one, hence its subtly different effect. According to personal taste, the flattening from tapping the neighbouring hole may be excessive and one can achieve a narrower fluctuation by tapping the next lower again open hole: for example, I use R3 to finger-vib a G - R2 is too broad a vibrato for me. Some cross fingerings can also be vibratoed by waving or tapping other fingers, which may involve a sharpening rather than a flattening of the main tone, e.g. on whistle, cross finger Cnat as oxx xox and vib with R1; on flute use oxo xxx and vib R1 also. With oxx ooo for C nat (if that is in tune enough to use), vibratoing R1 will flatten it, L3 will sharpen it.
Bottom D cannot be finger vibratoed in any way - even the slightest opening of the tonehole produces more of a trill effect. You can tap the end of the tube against your leg to partially obscure the end, or get someone to tap a finger on it for you or some such manoeuvre if you really want that effect on bottom D, or you can fall back on breath vib. You can wave/tap L1 on 2nd octave D, but I think this produces more of a tone-quality shift than a pitch change.
P.S. On a keyed flute, you can vibrato E with the Eb key,of course, if you wish.