I just graduated recently in May, and right before graduation, I asked if my professors could write me LOR for grad school. And they all agreed. Now I'm in the process of applying to grad schools, but it's almost October, and how should I write the email to remind my professors of LOR? Should I start by asking if their offers still stand, and if they need anything else (like supporting materials) from me?
You may do that. However, from May to October is not a very long time, so you could probably omit asking if the offer still stands. (But I suppose it doesn't do any harm.)
You can always include an unofficial transcript as an attachment. If they don't need that, they can ignore the attachment.