If the form doesn't work, you're probably best off emailing Support and giving them the information manually. I imagine you won't be the only person who encounters some kind of fluke issue, and I'm sure they'll take care of it =)
Viewing post in Unable to complete google form for free course
During the end-of-jam stream, Tim indicated that these emails would be sent out "the week of" June 10th, presumably because of the sheer number of submissions they need to verify. I don't really want to subvert this process by posting a link here because unfortunately, there are always going to be some cheaters who don't deserve their course, but if you feel at any point like you've been missed, skipped over, etc., the best thing to do is email GameDev Support and let them know.
No worries though, you have loads of time - the course doesn't need to be claimed until mid-July. Tim also indicated that he was very dedicated to ensuring nobody slips through the cracks regarding this, so it will get sorted one way or the other =)