I don't think it can do it alone. An Acrobat form can submit (via HTTP) to a server, which could take the data submitted by Acrobat, parse it, and submit the data to the USPS server. When it receives the response, it could then build a response that Acrobat will understand (FDF or XFDF) and return it. This intermediate server could be on the local machine.
To create PDF forms, you'd need Acrobat Pro or better, any version, Windows or Mac.