The dominant modern usage relates to excretion, but that nuance is irrelevant in the three places where Shakespeare uses the word.
When Romeo talks about the apothecary whom he plans to visit to buy poison, he describes his shop as containing “empty boxes, Green earthen pots, bladders” (Romeo and Juliet, V.i.46), by which he means simply “vessels derived from animals for storing liquid”.
And when Thersites talks of “the rotten diseases of the south... bladders full of impostume” (Troilus and Cressida, V.i.20, in the Quarto text only), he is referring generally to any cavity or vessel in the body.
The remaining usage, in Henry VIII, sometimes takes the reader by surprise.
This is when Wolsey says “I have ventured, Like little wanton boys that swim on bladders” (III.ii.359). Here the reference is simply to an air-filled bag which can be used in the manner of a lifejacket - what we would today call “water-wings”.