Hagrid is not shown casting a Patronus anywhere in the series, but still he was an Order member, and Order members usually communicate by means of a Patronus. Is there any indication that Hagrid was able to cast a Patronus, and if so, what was its form?
Hagrid could not produce a Patronus.
Hagrid couldn't produce a Patronus. It's a very difficult spell.
Since Hagrid was expelled before completing his studies at Hogwarts, it makes sense that - like many grown-up witches and wizards, in fact - he never learnt the difficult Patronus Charm. Some have also speculated that due to his unhappy childhood, he didn't have enough happy memories to draw on in order to cast it.