You can suggest a new banner since that would mean that you are requesting a change in the background picture, but if it is not your own personal club, as in you didn't create the club yourself, you 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 clubs - clubs with 100+ members, I believe. It was later expanded to clubs with 10+ members. Possibly, when the expansion was done, only icons and banners were added. The ability to choose a background image might still be only present in 100+ member clubs.