Process is an intransitive verb, but it means "walk in procession". Clergy and ministers process into a church service. It's pronounced differently, with the stress on the second syllable.
ODO doesn't allow process not to have an object, in the sense which is meant in your quote:
perform a series of mechanical or chemical operations on (something) in order to change or preserve it:
the salmon is quickly processed after harvest to preserve the flavour
* deal with (someone or something) using an official procedure:
the immigration authorities who processed him
* Computing operate on (data) by means of a program.
However, it is also fairly common for computer output messages to omit the verb, headline-style; or to use jargon.
In the quoted message, it is clear what is meant: the request went through the process. But this use is jargon.