Moodle already has exactly this feature! It's called the "Missing Word" format (look for the import button on the quiz category page). I'd not heard of the name "cloze" at the time, but a search on goodle.com seems to show it's the popular name for that format.
Actually, missing word (or short answer) is not a cloze task type. A classic cloze test is a longer passage with every nth word removed. The task is to recreate the text by putting in the missing words. However, this type is not very often used nowadays, and rightly so, because the words get removed at random without real pedagogical purpose. A more modern version of the cloze test is simply a text or passage with a number of words deleted. And this type can be very useful in teaching English as a foreign language, because it focuses on testing lexical and structural accuracy, common word collocations, etc. An example is in the attached zip file. Two main differences between Moodle's short-answer question and this cloze test are that 1/ there can be a lot of blanks in one question, and 2/ some blanks can have alternate (multiple) correct answers, e.g the third blank in the attached quiz can be answered with 'under', 'beneath' or 'underneath'.
A modified version of cloze test is a multiple-choice cloze, where for every gap there is a choice of answers to choose from (one correct and a few distractors). Again, an example in the attached zip file.
Neither of them is supported by Moodle at the moment. I have some experience with webct and they don't have any of these two question types either. However, it is possible to use their short-answer question in this way, because it allows to create a number of blanks for each question and specify which answer(s) can appear in which blank. It is not very elegant, because you get the text with numbered typed-in blanks (like this: (1)_____) which is then followed by the actual blanks (form fields), where the student writes the answers. Again, an example in the zip. However, I think this is not possible in Moodle at the moment.
With the multiple-choice cloze, there is a workaround in Moodle, though not perfect. You can create a number of multiple-choice questions and attach the text with blanks to the first one. However, the problem is that the whole set (all of these multiple-choice questions together with the text) should be treated as one question, and with this workaround it is treated as a number of multiple-choice questions.
These two question types (cloze and multiple-choice cloze) are definitely something I'd love to see implemented in Moodle, but knowing how busy Martin is I've never aksed for it, and I don't think I should On the other hand, I do not have the programming skills to do it myself or the financial resources to have it done by somebody else But who knows? Maybe in a few months?...
Just one final note: the tests in the attached files are not mine. They are taken from a sample of the Cambridge FCE exam available at http://www.cambridgeesol.org/support/dloads/ums.cfm