Can "process" be used as an intransitive verb?
Solution 1:
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:
verb
[with object]
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.
Solution 2:
The sentence
Your sentence processed successfully.
is a grammatical sentence. It contains an example of a derived intransitive verb: a verb used intransitively, but whose meaning is derived from the meaning of a basic transitive sense of the same verb. In this case, the change in transitivity is used to indicate what is usually called the middle voice.
The middle voice construction uses a transitive verb as if it were intransitive, with the intransitive subject corresponding to the object of the original transitive verb. It has a meaning similar to the passive voice. The process is quite common in English, and extends to a good number of verbs, though somewhat restricted in meaning. Verbs which participate in the middle voice construction are typically those whose meaning is more specific as to what the object is.
Some examples:
This cloth cuts easily.
This cloth folds easily.
Phenol and formaldehyde polymerize readily.
Bananas peel easily.
The program wouldn't run.
The script compiled successfully.
Other languages have a specific word-form for middle voice forms of a verb.