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The song Weird Fishes (Arpeggi) by Radiohead has a chord progression which has been shared in a website and I believe it is more or less correct:
The song seems to be in key B minor - which makes sense since it covers all chords used.
So the chord progression seems to be like iv-v-VII-VI and does not contain B minor at all (which would be i) while it is the key of the song.
Is this (a song being in a key and not using the tonic chord of that key at all) correct and does this happen often?
The key of the song tells you where the home note, tonic, is. When the harmony moves to the home chord, for example C major in the key of C major, it creates a sense of relief and being at rest. It's entirely possible to be aware of where home is, but still avoid going there, even indefinitely. That way you keep the listener feeling some tension and deny an easy solution or happy ending to the story.
You can try constructing such chord progressions yourself. Take an existing song that does go to the home chord, and substitute all those chords with something else.
Another interpretation of this song and chord progression is that is it not in a major or minor mode at all but in another mode, namely E dorian. The notes of the (modern) E dorian scale are E F# G A B C# D E, and all the chords you name can be found there; in addition, the keystone chord of the progression is an E chord, and the melody is plausibly anchored at E.
These alternate modes are not uncommon in popular music (mixolydian mode being the most common). Each mode can have a distinctive sound, just as major and minor do; and the dorian mode certainly can generate some restless feelings, with its chord progressions that resolve much less satisfyingly than in major, say.
As others have pointed out, it's perfectly possible to have music in which the tonic doesn't occur but we nevertheless get a clear sense of tonality. The opening of Beethoven's first symphony is a great example.
But to me the song "Weird Fishes" just doesn't seem to have a strong tonal center. I don't think it's in any particular key. This seems intentional. The song seems to be about death, loss of identity, and escaping worldly existence. They don't want it to sound like, "He's dead, he sank to the bottom of the ocean, and it's all over. Done, resolved." They're depicting death as a beginning, as a lack of resolution. If you had a clear cadence, resolving to a tonic and feeling stable there, it would be contrary to the effect that they're trying to produce.
Chord progression where tonic chord does not appear at all
The trick is in determining what the tonic (or implied tonic) is.
The traditional way to define the tonic is with dominant harmony (V I.) You could have a phrase that doesn't actually use a tonic chord, but if it ends with a convincing half cadence on the dominant, the tonic will be implied. Ex. vi IV ii V.
If you want the ambiguity of no tonic within a diatonic context (which is what Em7 F#m7 A A7 A6 Gmaj7 is, key signature of three sharps), then just play diatonic chords but avoid V to I.
A super common way to do that in a pop style is a vamp between relative major/minor, like Am to C. From those chords alone you can't tell which is tonic. Rhythm and melody could establish one of the other as "home" but strictly speaking the chords only can't. It could be vi I or i III.
Take this common progression D A Bm G - the Axis of Awesome - it could be nominally labelled I V vi IV, but it actually lacks a descending fifth progression (V I) and so harmonically it doesn't strongly define a tonic/dominant relationship. You could pick any of those four chords and treat it as home.
Progressions like this lots of repetition - a groove - because they don't contain a progression to form a final cadence. You could say it avoids functional harmony and frees you up to put chords it just about any order you like.
So, I would say "yes, it happens alot." Avoiding V I patterns is a device to eschew a clear tonic.
[C7 F][D7G]orE7 A7 D7 G7...a sequence often leads to a new key. Within the sequence there often isn't a tonic. In other words sequential harmony is tonally ambiguous by definition. The basic answer to the question is simply "yes", but I think the OP may be after something more meaningful like a whole song or section that seems to eschew a tonic.