I know that a pion can take place of an electron in an atom, but pion is not an elementary boson, but a particle made of constituents. However, my question regards elementary particles. Are there bound states for elementary bosons like photons, W, Z etc(maybe something like gluonium - two gluons bound together). And if yes, were they observed or is it only what theory allows?
In the spirit of "gluonium", there are (hypothetically) glueballs which can have far more than just two gluons. There isn't much else you can do.
You could also have "bound" W bosons, but for example a proton-W system would have the proton orbiting the much more massive W, not vice versa. A $W^+W^-$ system would be short-lived like positronium (assuming the Ws lived long enough for annihilation, but their short lifetime is a problem for any W-binding system).
Failing that you have to hope a weak or gravitational force can give you a bounded elementary boson, though again it may be short-lived.