What do the rules say ? "At 4th level, a druid gains the ability to turn herself into any small or Medium animal and back again once per day. Her options for new forms include all creatures with the animal type. This ability functions like the beast shape I spell, except as noted here."
The remainder doesn't say anything about armor, so let's look at Beast Shape I. Hmmm, doesn't mention armor, but it does tell us that it's a transmutation (polymorph).
Looking up polymorph ...
"When you cast a polymorph spell that changes you into a creature of the animal, dragon, elemental, magical beast, plant, or vermin type, all of your gear melds into your body. Items that provide constant bonuses and do not need to be activated continue to function while melded in this way (with the exception of armor and shield bonuses, which cease to function)."
No exceptions are mentioned for creatures that happen to be bipedal.
Drop your armor, wildshape, then don it again.
Or if you don't want to stay bipedal, get Wild armor. +3 bonus equivalent, but it lets armor work with wildshape.
Drop your armor, wildshape, then don it again
Or stay an ape all day, and just put your armor on at the beginning of the day most other characters have to. It is an option (well, by level 8 it is). You have a generally bipedal form, so all you really need is to take the feat that lets you talk while in animal form, and suddenly you have no mechanical problems being an ape all day once you get enough wildshape (roleplaying problems due to being a talking, magical ape of course still remain)
Admittedly, I tend to like this idea more with elementals, which even have the ability to speak. Earth elementals especially, due to the bonuses to strength and natural armor.
Drop your armor, wildshape, then don it again.
Depending on your normal race, I'm not sure the armor would fit (ignoring proportions etc, an ape is still a Large animal).
Armor would absolutely not fit while wild shaped (unless it was tailored to an ape, and therefore wouldn't fit a humanoid)
A minor but potentially important note. Armor with the Wild enhancement still melds into your body, but you retain the armor bonus. So either way you no longer suffer armor check penalty or max dex.
A lot of people seem to want to treat apes as bipedal they are not. As far as the game is concerned you need Harding for an ape. This means normal armour will meld.