Are you still a SAHM? If you are, I don't see why you couldn't just hang out in the morning or come a little early & see if you could meet the parents that way. Obviously if any of them/you work, this idea isn't feasible, but it seems like the easiest way to get the info you need.
Or, can't your DD just put the invite in their back pack in the morning, before school starts, or after? That way it's not done in front of the whole class. This is how DD received an invitation from her former classmate from last year.
Honestly, though, your DD is 7, almost 8, surely she can handle asking for their phone numbers, or giving them an invite. I don't think it's appropriate to involve an already swamped teacher in this.