The phrase "as quickly as we can" is correct and can be used in written English. You can use this phrase to mean that someone is trying to do something as quickly as possible or in the fastest way they can. For example: "We're working hard to finish the project as quickly as we can."
Let's negotiate as quickly as we can.
"We'll proceed as quickly as we can".
We're trying to fix that as quickly as we can.
"We're working as quickly as we can," she said.
"We are getting people out as quickly as we can.
"We are moving as quickly as we can," said Mnuchin.
Let's just get through the rest of the trailer as quickly as we can.
Will we get those in place as quickly as we can?
"We are trying to speak to the victims as quickly as we can.
We are actively investigating these reports and will provide information as quickly as we can.
We are doing everything we can to get him back as quickly as we can".
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union