I can’t tell from your post if you do the drop-off and/or pick-up for your son at school - if you do, that makes it undoubtedly easier, but it’s not impossible otherwise. The way it worked for me (my daughter has just finished her first year at school) involved a lot of lurking around, usually at pick up time. My daughter would identify someone she wanted to invite over, I would lurk a bit to spot the relevant parent, then I’d just approach them and introduce myself. It took a while before this happened - the first couple of months were just friendly hellos as we (the parents) passed each other at the school gates. I’d tend to be aware of the person as so-and-so’s mummy (or daddy or whatever) first of course, and approach them that way. You know, ‘hi, you’re Joe’s mum, aren’t you? I’m Charley, OldestDaugher’s mum’. From there you mention that the kids want to play together, fix up a date, exchange numbers etc. I find that an amused mention of the fact that ‘ha ha ha, I don’t even know your name! I only know you as Joe’s mum ha ha ha isn’t that funny?’ seems to break the ice, even though it’s the lamest, most predictable thing imaginable in the circumstances.
I am chronically shy with people I don’t know, and it’s hard to approach them, but I found that most people were in the same boat and at least we had the children in common. I wasn’t looking for friends, just other parents - although as it happened I’ve made some good friends this year.
Where either you or the other parent are not around after school, I found a note passed home with the other child works almost as well. Same lame joke, same subject matter etc. Here every child has a book bag that they use to transport stuff to and from school, and it’s quite common to have communications passed that way - I assume it’s similar where you are.
The contents of the playdate might be diferent where you are, but here (the UK) for children this age (5-6 this year) it’s almost always an after school till tea thing. Usually the hosting parent will pick my child up with their own, take them home and they will play and be fed. Either the hosting parent drops her off, or I pick her up from theirs, around 5.50-6.30 (school finishes at 3.15 for her). There have been a couple of weekend things, some after school trips as opposed to house playdates and so on, but they’ve been with children whose parents I’ve got to know, and never the first time. Some of the other parents are more cautious than I am, and have insisted on going over to the other party’s house in advance of the playdate to check it out, but I’ve never felt the need to do that. For this kind of playdate, in case it’s not clear, I’ve not been there. For arranged parties, I tend to stay - initially in case the whole atmosphere was a little overwhelming for my daughter, but for the last few months because it’s a nice opportunity to chat with the other parents and spend a couple of hours not attached to my younger daughter. We’ve also had a playdate with a younger girl (3) who we know from my childminder, where her mum did stay with her as she was a bit young to be left. We sat in the kitchen and drank tea whilst the girls disappeared upstairs.
That turned into a mammoth post Hope it was helpful, at least.