You're confusing two different things here. What do we do? is an interrogative sentence asking what course of action one should proceed with. The first do is one of those so-called auxiliary verbs in English that are used to form questions. The second do is nothing more than the main verb of the sentence. For example:
I've cleaned the room up, done the dishes and the laundry. What do I do now? Oh, I know what I'm going to do! I'm going to clean the toilet. I completely forgot about the toilet!
what we do, on the other hand, is a non-question statement—a noun clause, as Cardinal aptly pointed out in the comments section. It can only be properly understood as part of a larger context. In your case, where two organizations are telling you information about themselves, it simply means the following: this is what our sphere of activity is, this is what we specialize in or this is what our job is. The verb to do is commonly used to ask people questions about their jobs:
— What do you do for a living?
— I'm a taxi driver.
— What is your job?
— I do Photoshop designs.
You would say what we do on your website and then describe what exactly your job is to let visitors to your website know what you can offer them.