I've never fainted in therapy but I've fainted on multiple occassions, often due to standing in one spot for too long (it doesn't matter that my knees are bent and not locked) and being in a hot environment. I was eventually diagnosed with vasovagal syncope (extremely common among many people, especially women). One of the triggers is emotional distress or anxiety.

If it makes you feel any better, I have fainted on the first day of starting a new job about three times (three different jobs). I think those faints were due to anxiety and standing in one place for too long. It's horribly embarrassing but it's soon forgotten by those who witnessed it since I'm usually good at my jobs! I think your T has probably seen it all, you might be embarrassed now but you'll be able to brush it off in no time.