This is a very broad question, particularly since you put systemic in Systemic Functional Linguistics/Grammar in parentheses.
If you think about the broader functionalist program, you can hardly move within linguistics without encountering some aspect of it. You could say that most introduction to linguistics are written from the functionalist perspective. Most of the 'applied' disciplines of linguistics (with the notable exception of Natural Language Processing, Machine Translation and related fields) are broadly functionalist. From lexicography to language acquisition and language teaching. So are sociolinguistics and historical linguistics.
Things are a bit more murky when it comes to the Systemic Functional Linguistics. Again, you can see its direct impact in a number of fields. Most of the study of text-level and discourse-level phenomena is done (more or less explicitly) within the scope of the SFL approach. For example, Critical Discourse Analysis (as formulated, for instance, by Fairclough) explicitly relies on SFG. But you could also say that much of the work done in corpus linguistics is done under the umbrella of SFL. It would also be difficult to open a modern textbook of English as a second language and not see the impact of Haliday's work. Much of grammar writing is also done in that vein.
What makes things a bit more difficult is, that Systemic Functional Grammar is different from many of the other "Something Grammars" in that it does not come with a very constraining descriptive framework expressed through rules but rather a set of principles and axioms. This makes it a much more useful foundation for useful linguistic work and also slightly impervious to the sort of questions you might ask about generative-type grammars. It does not require constant tweaking of 'rules' and formal tools. As a linguist, you can just get on with things. It also does not require formal allegiance for it to be useful. I often refer to Halliday's three metafunctions of language in my own work.
As is so often the case, SFL is also subject to regional divisions. Most of the work on it is done in New Zealand and Australia (where Halliday relocated) with some remnants in the UK and around the world. It is therefore not as well known by many linguists who are actually feeling its impact in much of their daily work. Often in contrast to Chomsky whose work has almost no realistic impact on the work of a student of language but is widely known and cited. SFL has a similar problem in MAK Halliday whose stature perhaps makes people reluctant to declare their paradigmatic belonging to SFL for fear of being too constrained.
As to some exciting new ideas within SFL, I'm very intrigued by Michael Hoey's theory of lexical priming and John Sinclair's theory of local grammars.
As to further areas of research, here you're looking at all of language. Unlike the Chomskean paradigm, SFL does not constrain language to just what can be described through formal means. This makes it much more difficult to point to places with small gaps to fill. In my own work, I try to bring construction grammar and SFL closer together in the study of discourse level construction of arguments. But you can pick any area of language that interests you and keep SFG/SFL in mind as you work on it. You're certain to encounter some unexplored areas as you continue your research.
