With white keys only, namely a purely diatonic instrument, you are restricted to playing modal songs exclusively. For the Major mode, that is only an impediment with modulation or intermediate notes present. For church modes (of which strict Major mode is one), that works as well. For current-day uses of Minor mode (in its pure form also a church mode), you'll rarely make things fit well since there tend at least to be traces of Harmonic Minor in there, leaving the white-key tonality.
Folk songs are usually good material for going modal. Greensleeves works reasonably well in Dorian (white keys only starting with D) even though some phrases definitely shift character, but trying to pick a good mode for "Henry Martin" is trickier than it would look.
So generally, you'll likely end up with problems if your whole ensemble is bound to white keys: you are likely going to have to "backdate" songs uncompromisingly, giving them a flavor different from what they were written with. At least once you go beyond "Major mode".