I’m a bit incredulous at the other reactions to this question; but of course, you’ve A2A’d the right person.
You’re right, OP. bind and band and bundle all mean similar things. A band is something that you bind things with. An bundle is a bunch of things that have been bound together. Hey, bound is the past tense of bind! And for that matter, there’s also bond, which is a binding agreement. And as it turns out, bend as well (Reconstruction:Proto-Germanic/bandijaną), possibly because you bend a bow in order to bind it.
So if they’re all related, why do the vowels change?
Because Indo-European used ablaut to indicate various kinds of grammatical change. Ablaut involves vowel change in the stem, instead of using suffixes or prefixes to the stem. It is an old process, which is no longer productive; but you see it all over the place in several branches of Indo-European. You see it in the strong verbs of English: sing sang sung. You see it in the German stems underlying your three words: Proto-Germanic/bindaną, bandiz, bundą.
You’ll see it in Ancient Greek too. The related words temnō “I cut”, atomos “uncuttable”, at–mētos “uncut” are parallel to sing sang sung.