No, there can not be intentionality without an intentionalistic agent, nor can there be intending without an intender.
Anthropomorphizing rain and snow does not support the notion that because there can be snowing without a "snower", then there can be thinking without a thinker. The cause of snow and rain do not involve volition; these phenomena are the result of "brute" and non-intentionalistic cause. The description of snow having a "snower" is metaphorical.
Of note, intending is a kind of intentionality, but intentionality is not limited to intent. Reading your question, I am not sure if you are distinguishing intent and intentionality. Also of note, Brentano reintroduced the term and in intent German is "absicht", intend is "wollen" and intentional, "intentionale" and intentionality "intentionalität". Such are technical words English inherits from German by way of Latin...
All Descartes really proved is that thinking is happening, but there can be thinking without there being a thinker to do the thinking.
Is this what DesCartes said, or is this a dualistic inference?
As for a "neo-DesCartesian" proof of self which substitutes intentionality (or intending) for cogito, sure why not? Go for it. DesCartes, however, came to the conclusion of "cogito ergo sum" from a method of radical skepticism in which he doubted everything and discovered there was one thing he was unable to doubt: that he was thinking. He was not proving the existence of self, just that there is at the very least one thing which he could not doubt, i.e. his proof is epistemic, not ontological. Intendo ergo sum would be an interesting project, however, that you are intending to do something, you could still be deceived (if you deem the argument from hallucination convincing). It may be sufficient that your existence is demonstrated by your intention (or intentionlity) but is it necessary? Also, does thinking removed from a first person subjective ontology have a third-person objective or subjective or first-person objective ontology? Wouldn't these be contrary to the notion of intentionality?
If you are intending to also argue DesCartesian dualism, you might also benefit from this article, particularly section 5.