Ты can suggest a new banner since that would mean that Ты are requesting a change in the background picture, but if it is not your own personal club, as in Ты didn't create the club yourself, Ты cannot change the banner. Only the creator of the club is able to do that. I hope this helped.
I'm not certain of the reason why, but I have an idea. When the feature which allows us to vote for icons, banners and backgrounds first emerged, it was only present in larger клубы - клубы with 100+ members, I believe. It was later expanded to клубы with 10+ members. Possibly, when the expansion was done, only Иконки and banners were added. The ability to choose a background image might still be only present in 100+ member clubs.