You can use any WICED Wi-Fi platform to post to Twitter however there are a number of software modules that need to be written/ported.
Twitter uses OAuth for authentication however WICED currently does not provide an OAuth implementation in the SDK. There are a number of libraries readily available in source that can be ported to WICED however you will need to do this yourself.
On top of OAuth you will also need to implement the code to generate the REST queries as specified by the Twitter API.
I cannot comment on it's quality or completeness as I haven't looked at it in detail but it should give you a head start.
Porting an OAuth implementation will permit the WICED module to communicate with the Twitter servers directly and will not need a second party involved. An Arduino doesn't usually have the resources required to perform the cryptographic handshake and so the complex parts of the handshaking protocol are offloaded to another server (This depends on the hardware configuration of the Arduino).